christopher/oeis · Datasets at Hugging Face

a-number stringlengths 7 7	sequence sequencelengths 1 377	description stringlengths 3 852
A360768	[ "18", "24", "36", "48", "50", "54", "72", "75", "80", "90", "96", "98", "100", "108", "112", "120", "126", "135", "144", "147", "150", "160", "162", "168", "180", "189", "192", "196", "198", "200", "216", "224", "225", "234", "240", "242", "245", "250", "252", "264", "270", "288", "294", "300", "306", "312", "320", "324", "336", "338", "342", "350", "352", "360", "363", "375", "378", "384", "392", "396", "400", "405", "408" ]	Numbers k that are neither prime powers nor squarefree, such that k/rad(k) >= q, where rad(k) = A007947(k) and prime q = A119288(k).
A360769	[ "45", "63", "75", "99", "117", "135", "147", "153", "171", "175", "189", "207", "225", "245", "261", "275", "279", "297", "315", "325", "333", "351", "363", "369", "375", "387", "405", "423", "425", "441", "459", "475", "477", "495", "507", "513", "525", "531", "539", "549", "567", "575", "585", "603", "605", "621", "637", "639", "657", "675", "693", "711", "725", "735", "747", "765", "775", "783", "801", "819", "825" ]	Odd numbers that are neither prime powers nor squarefree.
A360770	[ "1", "5", "27", "260", "3125", "46684", "823543", "16777472", "387420498", "10000003125", "285311670611", "8916100495009", "302875106592253", "11112006826381559", "437893890380860625", "18446744073726328848", "827240261886336764177", "39346408075296925015353" ]	Expansion of Sum_{k>0} (x * (k + x^k))^k.
A360771	[ "1", "2", "5", "8", "20", "32", "77", "128", "288", "518", "1104", "2048", "4313", "8192", "16832", "32848", "66568", "131072", "264688", "524288", "1053737", "2097824", "4205568", "8388608", "16803744", "33554442", "67162112", "134222336", "268550704", "536870912", "1073999165", "2147483648", "4295493376", "8589962752" ]	Expansion of Sum_{k>=0} (x * (2 + x^k))^k.
A360772	[ "1", "2", "4", "5", "8", "9", "13", "16", "25", "32", "34", "64", "81", "89", "125", "128", "169", "233", "256", "441", "512", "610", "625", "729", "1024", "1156", "1597", "2048", "2197", "3025", "3125", "4096", "4181", "6561", "7921", "8192", "10946", "15625", "16384", "20736", "28561", "28657", "32768", "39304", "54289", "59049", "65536", "75025", "78125" ]	List of distinct numbers that are powers of odd-indexed Fibonacci numbers or even powers of nonzero even-indexed Fibonacci numbers.
A360773	[ "0", "1", "8", "1024", "620448" ]	Number of ways to tile a 2n X 2n square using rectangles with distinct dimensions such that the sum of the rectangles perimeters equals the area of the square.
A360774	[ "1", "1", "5", "31", "284", "3390", "49878", "871465", "17620450", "404554997", "10394845097", "295485704544", "9205957047661", "311922101632409", "11419004058232897", "449146827324857447", "18889836751306735360", "845892838094616177138", "40182354573647684880446" ]	Expansion of Sum_{k>=0} (x * (k + x))^k.
A360775	[ "1", "1", "4", "28", "260", "3152", "46913", "826677", "16823968", "388245283", "10016796672", "285699444297", "8926107792609", "303160590533808", "11120927427841820", "438196895219227683", "18457860168281435172", "827678295600605015006", "39364859979651634985089" ]	Expansion of Sum_{k>=0} (x * (k + x^2))^k.
A360776	[ "1", "1", "4", "27", "257", "3129", "46683", "823799", "16780342", "387467154", "10000823639", "285328449077", "8916487888186", "302885106945216", "11112292144568909", "437902806653498835", "18447046953316227905", "827251374022851280231", "39346845973273509115167" ]	Expansion of Sum_{k>=0} (x * (k + x^3))^k.
A360777	[ "9", "160", "143", "2679", "19933", "115248", "45", "1995" ]	a(n) is the index of the least n-gonal number that is the sum of two or more consecutive nonzero n-gonal numbers in more than one way, or -1 if no such number exists.
A360778	[ "1", "0", "0", "5", "4", "3", "2", "1", "0", "0", "2", "1", "0", "5", "4", "3", "2", "1", "0", "5", "4", "3", "2", "1", "0", "11", "10", "9", "8", "7", "6", "5", "4", "3", "2", "1", "0", "3", "2", "1", "0", "15", "14", "13", "12", "11", "10", "9", "8", "7", "6", "5", "4", "3", "2", "1", "0", "3", "2", "1", "0", "11", "10", "9", "8", "7", "6", "5", "4", "3", "2", "1", "0", "7", "6", "5", "4", "3", "2", "1", "0", "3", "2", "1", "0", "3", "2", "1", "0" ]	Smallest number k such that n + k is a refactorable number.
A360779	[ "1", "6", "1", "3", "6", "6", "12", "4", "16", "4", "12", "8", "4", "4", "8", "8", "4", "20", "4", "4", "16", "4", "24", "4", "20", "21", "3", "4", "8", "8", "4", "24", "12", "8", "32", "16", "4", "12", "12", "4", "8", "12", "28", "17", "3", "4", "2", "18", "4", "8", "8", "4", "12", "12", "20", "24", "4", "4", "16", "16", "12", "13", "7", "4", "4", "24", "8", "12", "24", "4", "8", "12", "44", "16", "12", "4", "16", "4", "24" ]	Refactorable numbers gaps: differences between consecutive refactorable numbers.
A360780	[ "1", "2", "8", "8", "8", "8", "8", "8", "9", "12", "12", "12", "18", "18", "18", "18", "18", "18", "24", "24", "24", "24", "24", "24", "36", "36", "36", "36", "36", "36", "36", "36", "36", "36", "36", "36", "40", "40", "40", "40", "56", "56", "56", "56", "56", "56", "56", "56", "56", "56", "56", "56", "56", "56", "56", "56", "60", "60", "60", "60", "72", "72", "72", "72", "72", "72", "72", "72", "72" ]	Least refactorable number > n.
A360781	[ "2", "3", "5", "7", "17", "19", "23", "29", "31", "37", "41", "43", "47", "53", "59", "61", "67", "71", "73", "79", "83", "101", "103", "107", "109", "113", "131", "139", "149", "151", "157", "163", "173", "179", "191", "193", "197", "211", "223", "227", "233", "239", "241", "251", "257", "263", "269", "271", "277", "281", "283", "293", "307", "311", "313", "317", "331" ]	Primes p such that at least one number remains prime when p is bracketed by a single digit d; that is, at least one instance of d//p//d is prime where // means concatenation.
A360782	[ "1", "1", "1", "3", "7", "16", "45", "125", "363", "1127", "3561", "11696", "39727", "138113", "494213", "1811075", "6784115", "25985928", "101520833", "404305549", "1640002039", "6767576175", "28395916893", "121048681024", "523902418555", "2300906314849", "10248029334297", "46266088140291" ]	Expansion of Sum_{k>=0} x^k / (1 - k*x^2)^(k+1).
A360783	[ "1", "1", "1", "1", "3", "7", "13", "24", "55", "133", "301", "678", "1639", "4120", "10253", "25591", "65869", "173551", "459493", "1225379", "3325123", "9162046", "25451181", "71296499", "202144225", "579612934", "1675822453", "4885178596", "14376297345", "42690792651", "127757371105", "385241085261", "1170960103855" ]	Expansion of Sum_{k>=0} x^k / (1 - k*x^3)^(k+1).
A360784	[ "1", "1", "3", "8", "18", "39", "86", "175", "352", "688", "1318", "2472", "4576", "8322", "14959", "26560", "46657", "81130", "139866", "239047", "405496", "682891", "1142466", "1899344", "3139432", "5160455", "8438871", "13732292", "22242647", "35867937", "57597730", "92121145", "146775205", "232998683", "368579188", "581091003" ]	Number of multisets of nonempty strict integer partitions with a total of n parts and total sum of 2n.
A360785	[ "1", "2", "5", "12", "26", "54", "112", "220", "427", "812", "1518", "2790", "5074", "9096", "16144", "28360", "49367", "85180", "145867", "247886", "418426", "701702", "1169673", "1938498", "3195497", "5240386", "8552308", "13892638", "22468406", "36184636", "58040397", "92737842", "147631545", "234184172", "370215442", "583343070" ]	Number of multisets of nonempty strict integer partitions with a total of 2n parts and total sum of 3n.
A360786	[ "0", "2", "42", "400", "2840", "17376", "97440", "516608", "2634624", "13058560", "63320576", "301707264", "1417009152", "6575120384", "30195425280", "137430827008", "620604391424", "2783097520128", "12403773407232", "54975376916480", "242441862512640", "1064326263734272", "4653131038195712", "20266193591992320" ]	Number of ways to place two dimers on an n-cube.
A360787	[ "1", "1", "1", "3", "13", "40", "177", "965", "4733", "28103", "184065", "1191888", "8713549", "67005689", "528870257", "4526024267", "40051790333", "368513578472", "3583302492545", "35868588067501", "373781214260749", "4052932682659599", "45218033687522481", "523234757502985824", "6245693941097387773" ]	Expansion of Sum_{k>=0} x^k / (1 - (k*x)^2)^(k+1).
A360788	[ "1", "1", "1", "1", "3", "25", "109", "324", "1135", "8803", "64189", "337854", "1707319", "13421410", "121248893", "894378619", "6082868725", "53046554917", "543432115477", "4989423130739", "42565774604131", "421544374075072", "4781440892689533", "51342685464272591", "522295380717090265" ]	Expansion of Sum_{k>=0} x^k / (1 - (k*x)^3)^(k+1).
A360789	[ "2", "3", "5", "7", "379", "23", "401", "61", "59", "29", "67", "71", "467", "79", "83", "179", "431", "89", "176557", "191", "24419", "491", "97", "101", "499", "1213", "3169", "3191", "523", "229", "3187", "223", "3203", "8609", "3163", "251", "176509", "257", "24509", "263", "3253", "269", "547", "3347", "1304867", "293" ]	Least prime p such that p mod primepi(p) = n.
A360790	[ "8", "13", "41", "53", "137", "173", "305", "397", "533", "877", "977", "1373", "1697", "1885", "2245", "2813", "3517", "3737", "4493", "5077", "5345", "6277", "6953", "7937", "9413", "10217", "10613", "11465", "12077", "12785", "16165", "17165", "18869", "19325", "22237", "22837", "24665", "26605", "27925", "29933", "32141", "32765", "36497", "37253", "38953", "39745" ]	Squared length of diagonal of right trapezoid with three consecutive prime length sides.
A360791	[ "1", "2", "4", "14", "28", "94", "218", "588", "1366", "3618", "8134", "20320", "45592", "105810", "236960", "539392", "1174530", "2612436", "5628606", "12226350", "26130568", "55938126", "117997774", "249680514", "523032956", "1094500962", "2275886514", "4727461792", "9762182762", "20148991512", "41403646304", "84961079990" ]	Sum of all prime encoded complete partitions of n.
A360792	[ "0", "0", "1", "2", "3", "4", "5", "7", "9", "10", "12", "15", "17", "19", "22", "25", "28", "31", "34", "37", "41", "45", "49", "53", "57", "61", "66", "71", "75", "80", "86", "91", "96", "102", "108", "114", "120", "126", "133", "139", "146", "153", "160", "167", "175", "182", "190", "198", "206", "214", "223", "231", "240", "249", "258", "267", "276", "285", "295", "305" ]	Integer portion of area of inscribed circle in a regular polygon having n sides of unit length.
A360793	[ "24", "40", "54", "56", "88", "104", "120", "135", "136", "152", "168", "184", "189", "232", "248", "250", "264", "270", "280", "296", "297", "312", "328", "344", "351", "375", "376", "378", "408", "424", "440", "456", "459", "472", "488", "513", "520", "536", "552", "568", "584", "594", "616", "621", "632", "664", "680", "686", "696", "702", "712", "728", "744", "750" ]	Numbers of the form m*p^3, where m > 1 is squarefree and prime p does not divide m.
A360794	[ "1", "3", "4", "11", "6", "43", "8", "109", "100", "281", "12", "1507", "14", "1863", "3376", "6937", "18", "26245", "20", "53211", "63022", "67739", "24", "572413", "78776", "372945", "1087048", "1761719", "30", "7362871", "32", "9947953", "16897486", "10027349", "8011116", "123101515", "38", "49807779", "241823440", "361722421", "42" ]	Expansion of Sum_{k>0} x^k / (1 - k * x^k)^(k+1).
A360795	[ "1", "3", "4", "17", "6", "211", "8", "1929", "7300", "22601", "12", "1724809", "14", "6703047", "223678576", "738787345", "18", "65630598229", "20", "2119646503661", "24448573943662", "3423809253371", "24", "21453113652593665", "12016296386718776", "4240253019018225", "8255251542208471048", "67251293544533119589", "30" ]	Expansion of Sum_{k>0} x^k / (1 - (k * x)^k)^(k+1).
A360796	[ "7", "9", "11", "13", "14", "17", "17", "19", "20", "25", "23", "29", "26", "27", "29", "37", "31", "40", "34", "35", "38", "46", "39", "41", "44", "43", "44", "54", "47", "58", "49", "51", "56", "53", "54", "67", "62", "59", "59", "70", "62", "73", "64", "65", "74", "78", "69", "71", "71", "75", "74", "86", "76", "77", "79", "83", "92", "93", "83", "103" ]	a(n) > n is the smallest integer such that there exist integers n < c <= d < a(n) satisfying n^2 + a(n)^2 = c^2 + d^2.
A360797	[ "1", "5", "13", "39", "81", "225", "449", "1115", "2345", "5373", "11265", "25483", "53249", "116497", "246405", "529195", "1114113", "2372741", "4980737", "10515511", "22025617", "46204953", "96468993", "201506607", "419432417", "872787997", "1811981789", "3758970975", "7784628225", "16108217801", "33285996545", "68723976779" ]	Expansion of Sum_{k>0} x^k / (1 - 2 * x^k)^(k+1).
A360798	[ "1", "5", "13", "45", "81", "321", "449", "1745", "2945", "9153", "11265", "60609", "53249", "230401", "410625", "1259777", "1114113", "7263233", "4980737", "31337473", "44630017", "115367937", "96468993", "937283585", "551550977", "2399256577", "4594597889", "14579646465", "7784628225", "89894944769", "33285996545" ]	Expansion of Sum_{k>0} x^k / (1 - (2 * x)^k)^(k+1).
A360801	[ "1", "3", "5", "13", "17", "51", "65", "169", "281", "603", "1025", "2373", "4097", "8655", "16685", "33969", "65537", "134151", "262145", "530269", "1050481", "2108439", "4194305", "8420201", "16778337", "33607707", "67120565", "134338493", "268435457", "537151131", "1073741825", "2148024289", "4295035145", "8591048739" ]	Expansion of Sum_{k>0} (x / (1 - 2 * x^k))^k.
A360802	[ "1", "3", "5", "17", "17", "105", "65", "449", "641", "1953", "1025", "16257", "4097", "37761", "93185", "247809", "65537", "1499649", "262145", "6596609", "8847361", "13654017", "4194305", "210026497", "90177537", "251764737", "833880065", "2659418113", "268435457", "18345328641", "1073741825", "53553922049", "75438751745" ]	Expansion of Sum_{k>0} (x / (1 - (2 * x)^k))^k.
A360803	[ "3", "313", "94863", "298327", "987917", "3162083", "9893887", "29983327", "99477133", "99483667", "197483417", "282753937", "314623583", "315432874", "706399164", "773303937", "894303633", "947047833", "948675387", "989938887", "994927133", "994987437", "998398167", "2428989417", "2754991833", "2983284917", "2999833327" ]	Numbers whose squares have a digit average of 8 or more.
A360804	[ "1", "1", "21", "253", "2401", "36237", "815929", "18713197" ]	Number of ways to tile an n X n square using rectangles with distinct areas.
A360805	[ "0", "31", "120", "283", "293", "712", "2872", "3287", "5028", "5129", "7088", "9553", "13229", "14232", "14799", "15113", "20153", "20830", "23239", "30233", "31430", "31667", "34443", "40654", "44298", "50184", "78877", "105834", "115281", "125120", "164253", "192103", "201590", "227747", "239910", "241910", "282230", "322550", "374370" ]	Nonnegative integers k such that k! mod nextprime(k) is larger than k.
A360806	[ "1", "2", "8", "80", "2240", "215040", "77414400", "61931520000", "170930995200000", "1340099002368000000" ]	a(0) = 1; for n >= 1, a(n) is the least integer k > a(n-1) such that k / A000005(k) = a(n-1).
A360808	[ "1", "2", "2", "2", "8", "8", "35", "16", "51", "145", "1112", "1145", "10929", "41400", "542785", "40384", "583169", "2781808", "48558706", "65461347", "1277941540", "7370563251", "159694747220", "63387056365", "1500631724572", "10152855622657" ]	Number of double cosets of the Sylow 2-subgroup of the symmetric group S_n.
A360809	[ "2", "5", "8", "6", "7", "0", "5", "0", "5", "9", "7", "8", "6", "8", "0", "8", "2", "2", "7", "7", "7", "8", "1", "0", "6", "8", "7", "2", "9", "4", "6", "9", "6", "0", "2", "1", "3", "5", "7", "3", "0", "9", "6", "2", "7", "4", "2", "4", "8", "9", "3", "6", "1", "2", "4", "4", "6", "7", "0", "8", "2", "4", "2", "2", "5", "8", "5", "9", "4", "0", "4", "5", "5", "6", "0", "6", "6", "4", "3", "4", "2", "6", "4", "2", "8", "8", "2", "7", "7", "7", "5", "6", "7", "5", "3", "9", "0", "8", "8", "7", "6", "4", "4", "6", "9", "9", "8", "1" ]	Decimal expansion of the area under the curve of the reciprocal of the Luschny factorial function from zero to infinity.
A360810	[ "1", "1", "1", "2", "5", "11", "29", "81", "229", "696", "2181", "7045", "23653", "81433", "288173", "1046814", "3887749", "14768783", "57275541", "226462801", "912443397", "3741515804", "15603500797", "66134448329", "284660214181", "1243605590897", "5511058189989", "24760003963802", "112726590916645" ]	Expansion of Sum_{k>=0} ( x / (1 - k * x^2) )^k.
A360811	[ "1", "1", "1", "1", "2", "5", "10", "18", "38", "91", "211", "472", "1108", "2754", "6881", "17101", "43443", "113565", "300142", "797191", "2147414", "5883976", "16293712", "45471429", "128285353", "366266188", "1055534118", "3066483484", "8989837397", "26602652605", "79370560477", "238606427241", "722973445270" ]	Expansion of Sum_{k>=0} ( x / (1 - k * x^3) )^k.
A360812	[ "1", "1", "1", "2", "9", "29", "113", "613", "3033", "17010", "110929", "713249", "5061097", "38762873", "302389553", "2544613578", "22404995001", "203762678941", "1960880744337", "19509713674397", "201306862742217", "2166901479447194", "24018963506471921", "275731857268608673", "3271769647891351705" ]	Expansion of Sum_{k>=0} ( x / (1 - (k * x)^2) )^k.
A360813	[ "1", "1", "1", "1", "2", "17", "82", "258", "818", "5671", "43363", "240520", "1183168", "8547054", "77831681", "596258173", "4031934111", "33313129161", "338733239446", "3187239159511", "27197807726066", "260179611473044", "2918973182685904", "31820249821418229", "324099587971865989" ]	Expansion of Sum_{k>=0} ( x / (1 - (k * x)^3) )^k.
A360814	[ "1", "0", "1", "2", "4", "10", "30", "98", "338", "1240", "4877", "20496", "91213", "426678", "2090081", "10702438", "57193760", "318283388", "1840036058", "11026424446", "68370955450", "438039068726", "2896018310881", "19733372875632", "138418266287689", "998363508783924", "7396739279819185", "56239695790595786" ]	Expansion of Sum_{k>=0} x^(2k) / (1 - kx)^(k+1).
A360815	[ "1", "0", "0", "1", "2", "3", "5", "11", "30", "88", "260", "771", "2343", "7474", "25380", "91650", "347988", "1371873", "5570173", "23233703", "99676434", "440931977", "2014619700", "9506385864", "46246356169", "231348803925", "1187212953132", "6239006165820", "33546182775824", "184497923546700" ]	Expansion of Sum_{k>=0} x^(3k) / (1 - kx)^(k+1).
A360816	[ "1", "0", "1", "2", "19", "100", "1118", "10034", "134993", "1715140", "27589661", "449763360", "8522965956", "168431719308", "3698624353289", "85523954588806", "2142927489388319", "56618555339223572", "1596938935380604858", "47399670488829289678", "1487559109670284821841" ]	Expansion of Sum_{k>=0} (kx)^(2k) / (1 - k*x)^(k+1).
A360817	[ "1", "0", "0", "1", "2", "3", "68", "389", "1542", "24810", "251564", "1814487", "27520734", "391640548", "4295115396", "69305652406", "1221344986380", "18207710383335", "329699350020676", "6759819628538561", "126950556666301050", "2624697847966227077", "60825028694289947940", "1365568620213461601924" ]	Expansion of Sum_{k>=0} (kx)^(3k) / (1 - k*x)^(k+1).
A360818	[ "1", "0", "1", "1", "17", "65", "922", "7074", "106183", "1248479", "21144289", "331763177", "6441011484", "124904347404", "2773880604749", "63538143151589", "1600211849569585", "42076439530000297", "1189408501356380558", "35214128238218917974", "1106088535644470694779" ]	Expansion of Sum_{k>=0} ( (kx)^2 / (1 - kx) )^k.
A360819	[ "1", "0", "0", "1", "1", "1", "65", "257", "769", "21732", "182268", "1075171", "22120299", "292415838", "2784944366", "52394511682", "914813711338", "12411977351379", "240868108545883", "5024364548461861", "88977315031536205", "1888119425325238979", "44744897995532996819", "971263427084750362992" ]	Expansion of Sum_{k>=0} ( (kx)^3 / (1 - kx) )^k.
A360820	[ "1", "4", "48", "1792", "221184", "98566144", "173946175488", "1281755680079872", "39534286378918477824", "5018464395368794081460224", "2586745980900067184722499862528", "5375203895735606878055792019528220672", "44865714160227204455469409035569750630989824", "1501355804811017489524770237231795462175548447391744" ]	a(n) = Sum_{k=0..n} binomial(n, k)2^(n^2-k(n-k)).
A360821	[ "0", "1", "1", "1", "1", "1", "0", "1", "1", "2", "1", "2", "2", "2", "2", "2", "2", "2", "3", "3", "2", "2", "2", "2", "1", "2", "1", "1", "2", "2", "2", "2", "2", "2", "2", "2", "1", "1", "2", "3", "2", "2", "2", "2", "2", "2", "3", "3", "3", "3", "3", "3", "4", "4", "3", "3", "2", "2", "2", "3", "3", "3", "3", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "3", "3", "3", "4", "4", "4", "4", "4", "5", "5", "5", "5", "5", "6" ]	Number of primes of the form k^2+1 between n^2 and 2*n^2 exclusive.
A360822	[ "1", "2", "3", "4", "5", "6", "7", "8", "9", "13", "14", "17", "22", "23", "27", "28", "29", "30", "31", "33", "43", "53", "63", "67", "77", "83", "91", "93", "94", "97", "99", "141", "167", "173", "197", "283", "293", "297", "298", "303", "313", "314", "316", "447", "583", "707", "767", "833", "836", "917", "943", "947", "1378", "2917", "2983", "3033", "5467", "9417", "9433", "29983", "31367", "94863" ]	Numbers whose squares have at most 2 digits less than 8.
A360823	[ "1", "4", "6", "20", "10", "96", "14", "256", "288", "650", "22", "4200", "26", "4004", "11160", "18784", "34", "70758", "38", "164140", "196098", "136664", "46", "1756728", "393800", "747890", "3287844", "5452076", "58", "22563060", "62", "31220032", "50767926", "20059286", "41640130", "391194396", "74", "99622016", "725647728", "1298396440" ]	Expansion of Sum_{k>0} k * x^k / (1 - k * x^k)^(k+1).
A360824	[ "1", "6", "30", "284", "3130", "47082", "823550", "16782664", "387422928", "10000094720", "285311670622", "8916102486528", "302875106592266", "11112006871683606", "437893890382576560", "18446744074918103056", "827240261886336764194", "39346408075331452862196" ]	Expansion of Sum_{k>0} (k * x)^k / (1 - k * x^k)^(k+1).
A360825	[ "1", "1", "2", "2", "4", "1", "6", "2", "5", "1", "10", "1", "12", "3", "8", "1", "16", "1", "18", "4", "11", "1", "22", "22", "6", "5", "14", "1", "28", "1", "30", "33", "20", "31", "18", "1", "36", "7", "20", "1", "40", "1", "42", "8", "23", "1", "46", "19", "11", "9", "26", "1", "52", "30", "27", "10", "29", "1", "58", "1", "60", "43", "53", "56", "33", "1", "66", "12", "35", "1", "70", "1", "72", "27", "23" ]	a(n) is the remainder after dividing n! by its least nondivisor.
A360827	[ "443", "647", "1847", "2243", "2687", "2699", "6263", "6563", "7487", "7583", "8627", "8663", "9419", "9767", "10223", "11867", "12323", "13187", "13907", "14627", "14723", "14783", "17747", "17783", "19739", "20639", "20807", "21863", "22307", "23747", "24107", "24923", "25127", "26759", "27983", "29207", "29819", "30839", "31247", "32303", "34403", "34439" ]	Primes p, not safe primes, such that the smallest factor of (2^(p-1)-1) / 3 is equal to p.
A360828	[ "1", "6", "6", "4", "8", "2", "2", "8", "4", "2", "9", "7", "8", "3", "3", "9", "3", "9", "4", "0", "0", "3", "0", "9", "5", "7", "2", "8", "2", "9", "0", "6", "5", "6", "9", "8", "1", "7", "4", "3", "0", "2", "2", "8", "5", "8", "6", "1", "4", "0", "9", "9", "6", "8", "9", "6", "4", "7", "1", "0", "8", "3", "2", "2", "7", "3", "6", "5", "6", "3", "9", "4", "5", "6", "3", "5", "4", "8", "6", "3", "2", "3", "6", "3", "0", "9", "2", "7", "3", "3", "4", "6", "1", "8", "3", "7", "2", "2", "9", "4" ]	Decimal expansion of the ratio between the perimeter of the first Morley triangle of an isosceles right triangle and the perimeter of this isosceles right triangle.
A360829	[ "3", "1", "0", "8", "8", "9", "1", "3", "2", "4", "5", "5", "3", "5", "2", "6", "3", "6", "7", "3", "0", "3", "1", "0", "9", "7", "6", "3", "5", "2", "7", "6", "6", "4", "2", "1", "4", "9", "9", "0", "9", "1", "9", "4", "1", "6", "8", "1", "6", "6", "0", "9", "9", "0", "9", "7", "6", "6", "2", "2", "1", "4", "0", "4", "0", "8", "8", "2", "7", "7", "9", "5", "9", "0", "4", "0", "0", "0", "6", "4", "8", "9", "2", "0", "0", "5", "8", "2", "6", "8", "2", "5", "1", "8", "5", "0", "0", "8" ]	Decimal expansion of the ratio between the area of the first Morley triangle of an isosceles right triangle and its area.
A360830	[ "1", "3", "6", "42", "84", "252", "2772", "36036", "612612", "11639628", "267711444", "803134332", "23290895628", "722017764468", "1444035528936", "53429314570632", "2190601897395912", "94195881588024216", "4427206434637138152", "30990445042459967064" ]	Numbers that when concatenated with the natural numbers from 1 to N are divisible by the corresponding order number.
A360831	[ "1", "6", "30", "308", "3130", "49962", "823550", "17107464", "387617328", "10058609120", "285311670622", "8931600297696", "302875106592266", "11117432610599574", "437894531752211760", "18449277498826162192", "827240261886336764194", "39347911865350001626164" ]	Expansion of Sum_{k>0} (k * x)^k / (1 - (k * x)^k)^(k+1).
A360832	[ "1", "1", "4", "28", "288", "3855", "63232", "1227291", "27511296", "699389444", "19880700928", "624817997477", "21512488648704", "805233062024021", "32556682898653184", "1413981749074790444", "65652661019642560512", "3245240681196968168619", "170146759140135777861632" ]	Expansion of Sum_{k>=0} ( k * x / (1 - (k * x)^2) )^k.
A360833	[ "1", "1", "4", "27", "257", "3189", "48843", "889080", "18731109", "448004763", "11987812504", "354763577414", "11503684020051", "405589341060930", "15447798292502206", "632069580794524857", "27649951709582591394", "1287748889361331630661", "63616184683123273364961" ]	Expansion of Sum_{k>=0} ( k * x / (1 - (k * x)^3) )^k.
A360834	[ "1", "1", "4", "29", "304", "4100", "67520", "1314167", "29520128", "751658635", "21393444864", "673046604600", "23192501108736", "868730852002205", "35145114836811776", "1527192185786650417", "70941146068492943360", "3508043437942077557884", "183989995827118805352448" ]	Expansion of Sum_{k>=0} (k * x)^k / (1 - (k * x)^2)^(k+1).
A360835	[ "1", "1", "4", "27", "258", "3221", "49572", "905466", "19122502", "458161191", "12275530636", "363646493044", "11801356347294", "416365459777150", "15867258718677348", "649548679156603983", "28426564854590132236", "1324406974148881529057", "65448443631801436742052" ]	Expansion of Sum_{k>=0} (k * x)^k / (1 - (k * x)^3)^(k+1).
A360836	[ "12880", "18896570" ]	a(n) is the least n-gonal pyramidal number that is the sum of two or more consecutive nonzero n-gonal pyramidal numbers in more than one way.
A360837	[ "1", "3", "59", "10079", "744666", "163710521" ]	a(n) is the least positive integer that can be expressed as the sum of one or more consecutive prime-indexed primes in exactly n ways.
A360839	[ "1", "6", "32", "103", "250", "220" ]	Number of minimal graphs of twin-width 2 on n unlabeled vertices.
A360840	[ "432", "2592", "139968", "444528", "472392", "995328", "3456000", "5174928", "6561000", "10125000", "15552000", "15804072", "17496000", "25299648", "28449792", "37340352", "54675000", "63700992", "85957848", "88723728", "99574272", "120891312", "144027072", "169869312", "177147000", "197413632", "253125000", "259308000" ]	3-full numbers (A036966) sandwiched between twin primes.
A360841	[ "2592", "139968", "995328", "37340352", "63700992", "99574272", "169869312", "414720000", "1399680000", "4076863488", "10871635968", "17714700000", "22781250000", "25312500000", "35888419872", "51840000000", "82012500000", "98802571392", "135304020000", "136136700000", "170749797552", "174960000000", "196730062848" ]	4-full numbers (A036967) sandwiched between twin primes.
A360842	[ "139968", "995328", "63700992", "4076863488", "17714700000", "82012500000", "98802571392", "174960000000", "445240556352", "641194278912", "889223142528", "1059917571072", "1594323000000", "1663012435968", "2348273369088", "3333709317312", "5717741400000", "16260080320512", "19144761127488", "28697814000000" ]	5-full numbers (A069492) sandwiched between twin primes.
A360843	[ "139968", "98802571392", "174960000000", "889223142528", "1594323000000", "2348273369088", "19144761127488", "28697814000000", "56358560858112", "84537841287168", "150289495621632", "186624000000000", "328341017826432", "369056250000000", "392147405854848", "578415690713088", "597871125000000" ]	6-full numbers (A069493) sandwiched between twin primes.
A360844	[ "4", "432", "2592", "139968", "139968", "174960000000", "56358560858112", "84537841287168", "578415690713088", "578415690713088", "1141260857376768", "61628086298345472", "61628086298345472", "61628086298345472", "322850407500000000000000000000", "322850407500000000000000000000", "62518864539857068333550694039552" ]	a(n) is the least k-full number that is sandwiched between twin primes.
A360846	[ "1", "3", "3", "4", "8", "4", "4", "17", "17", "4", "4", "32", "65", "32", "4", "4", "66", "222", "222", "66", "4", "4", "130", "766", "1280", "766", "130", "4", "4", "262", "2685", "7629", "7629", "2685", "262", "4", "4", "522", "9450", "46032", "78981", "46032", "9450", "522", "4", "4", "1046", "33158", "278419", "820308", "820308", "278419", "33158", "1046", "4" ]	Array read by antidiagonals: T(m,n) is the number of dominating induced trees in the grid graph P_m X P_n.
A360847	[ "1", "8", "65", "1280", "78981", "14605388", "7904828158", "12456744197696", "57118103869618858", "760896261783236975004", "29416443122724544970455433", "3297715940113139272931793598648", "1071333966021766251746119497973623975", "1008129126269380724757869194465038817386728" ]	Number of dominating induced trees in the n X n grid graph.
A360848	[ "3", "8", "17", "32", "66", "130", "262", "522", "1046", "2090", "4182", "8362", "16726", "33450", "66902", "133802", "267606", "535210", "1070422", "2140842", "4281686", "8563370", "17126742", "34253482", "68506966", "137013930", "274027862", "548055722", "1096111446", "2192222890", "4384445782", "8768891562" ]	Number of dominating induced trees in the n-ladder graph P_2_X P_n.
A360849	[ "0", "0", "0", "0", "1", "0", "0", "3", "3", "0", "0", "6", "15", "6", "0", "0", "10", "42", "42", "10", "0", "0", "15", "90", "204", "90", "15", "0", "0", "21", "165", "660", "660", "165", "21", "0", "0", "28", "273", "1650", "3940", "1650", "273", "28", "0", "0", "36", "420", "3486", "15390", "15390", "3486", "420", "36", "0", "0", "45", "612", "6552", "45150", "113865", "45150", "6552", "612", "45", "0" ]	Array read by antidiagonals: T(m,n) is the number of (undirected) cycles in the complete bipartite graph K_{m,n}.
A360850	[ "1", "3", "3", "6", "12", "6", "10", "33", "33", "10", "15", "72", "135", "72", "15", "21", "135", "438", "438", "135", "21", "28", "228", "1140", "2224", "1140", "228", "28", "36", "357", "2511", "8850", "8850", "2511", "357", "36", "45", "528", "4893", "27480", "55725", "27480", "4893", "528", "45", "55", "747", "8700", "70462", "265665", "265665", "70462", "8700", "747", "55" ]	Array read by antidiagonals: T(m,n) is the number of (undirected) paths in the complete bipartite graph K_{m,n}.
A360851	[ "0", "1", "1", "3", "8", "3", "6", "27", "27", "6", "10", "64", "126", "64", "10", "15", "125", "426", "426", "125", "15", "21", "216", "1125", "2208", "1125", "216", "21", "28", "343", "2493", "8830", "8830", "2493", "343", "28", "36", "512", "4872", "27456", "55700", "27456", "4872", "512", "36", "45", "729", "8676", "70434", "265635", "265635", "70434", "8676", "729", "45" ]	Array read by antidiagonals: T(m,n) is the number of induced paths in the rook graph K_m X K_n.
A360852	[ "0", "8", "126", "2208", "55700", "2006280", "98309778", "6291829376", "509638185288", "50963818537800", "6166622043087110", "887993574204562848", "150070914040571147676", "29413899151951944980168", "6618127309189187620585050", "1694240591152432030869834240", "489635530843052856921382173968" ]	Number of induced paths in the n X n rook graph.
A360853	[ "0", "0", "0", "1", "1", "1", "4", "5", "5", "4", "10", "14", "21", "14", "10", "20", "30", "58", "58", "30", "20", "35", "55", "125", "236", "125", "55", "35", "56", "91", "231", "720", "720", "231", "91", "56", "84", "140", "385", "1754", "4040", "1754", "385", "140", "84", "120", "204", "596", "3654", "15550", "15550", "3654", "596", "204", "120" ]	Array read by antidiagonals: T(m,n) is the number of induced cycles in the rook graph K_m X K_n.
A360854	[ "0", "1", "21", "236", "4040", "114105", "4662721", "256485936", "18226110456", "1623855703785", "177195820502965", "23237493232958796", "3605437233380103056", "653193551573628910481", "136634950180317224879985", "32681589590709963123110080", "8863149183726257535369656976" ]	Number of induced cycles in the n X n rook graph.
A360855	[ "0", "0", "0", "1", "0", "1", "4", "2", "2", "4", "10", "8", "6", "8", "10", "20", "20", "16", "16", "20", "20", "35", "40", "35", "32", "35", "40", "35", "56", "70", "66", "60", "60", "66", "70", "56", "84", "112", "112", "104", "100", "104", "112", "112", "84", "120", "168", "176", "168", "160", "160", "168", "176", "168", "120", "165", "240", "261", "256", "245", "240", "245", "256", "261", "240", "165" ]	Array read by antidiagonals: T(m,n) is the number of triangles in the rook graph K_m X K_n.
A360856	[ "1", "1", "2", "6", "16", "48", "140", "424", "1280", "3920", "12032", "37184", "115248", "358624", "1118784", "3499584", "10969344", "34450944", "108377984", "341465344", "1077300224", "3403006464", "10761447424", "34065967104", "107937899264", "342293526016", "1086339120128", "3450236511232", "10965437349888" ]	a(n) = [x^n](1/2)(1 + (2x + 1)/sqrt(1 - 8x^2(x + 1))).
A360857	[ "1", "1", "1", "1", "2", "6", "1", "3", "12", "12", "1", "4", "20", "30", "60", "1", "5", "30", "60", "150", "150", "1", "6", "42", "105", "315", "420", "700", "1", "7", "56", "168", "588", "980", "1960", "1960", "1", "8", "72", "252", "1008", "2016", "4704", "5880", "8820", "1", "9", "90", "360", "1620", "3780", "10080", "15120", "26460", "26460" ]	Triangle read by rows. T(n, k) = binomial(n, ceil(k/2)) * binomial(n + 1, floor(k/2)).
A360858	[ "1", "1", "2", "1", "3", "6", "1", "4", "12", "18", "1", "5", "20", "40", "60", "1", "6", "30", "75", "150", "200", "1", "7", "42", "126", "315", "525", "700", "1", "8", "56", "196", "588", "1176", "1960", "2450", "1", "9", "72", "288", "1008", "2352", "4704", "7056", "8820", "1", "10", "90", "405", "1620", "4320", "10080", "17640", "26460", "31752" ]	Triangle read by rows. T(n, k) = binomial(n + 1, ceil(k/2)) * binomial(n, floor(k/2)).
A360859	[ "1", "1", "1", "1", "2", "4", "1", "3", "9", "9", "1", "4", "16", "24", "36", "1", "5", "25", "50", "100", "100", "1", "6", "36", "90", "225", "300", "400", "1", "7", "49", "147", "441", "735", "1225", "1225", "1", "8", "64", "224", "784", "1568", "3136", "3920", "4900", "1", "9", "81", "324", "1296", "3024", "7056", "10584", "15876", "15876", "1", "10", "100", "450", "2025", "5400", "14400", "25200", "44100", "52920", "63504" ]	Triangle read by rows. T(n, k) = binomial(n, ceil(k/2)) * binomial(n, floor(k/2)).
A360860	[ "1", "0", "1", "0", "1", "2", "0", "2", "4", "8", "0", "8", "14", "26", "64", "0", "64", "96", "144", "296", "1024", "0", "1024", "1344", "1664", "2424", "6064", "32768", "0", "32768", "38912", "42752", "48832", "70672", "230896", "2097152", "0", "2097152", "2326528", "2412544", "2497664", "2701504", "3823072", "16886864", "268435456" ]	Accumulation triangle of A360603 read by rows.
A360861	[ "1", "2", "7", "22", "81", "281", "1058", "3830", "14605", "54127", "208110", "782761", "3027038", "11501478", "44668692", "170974710", "666220005", "2564271875", "10018268150", "38728479647", "151631858378", "588229029258", "2307174835212", "8975958379817", "35258881445606", "137501193282026", "540821096592028" ]	a(n) = Sum_{k=0..n} binomial(n, ceil(k/2)) * binomial(n, floor(k/2)).
A360862	[ "1", "1", "2", "1", "4", "1", "7", "5", "1", "10", "20", "5", "1", "14", "48", "36", "1", "18", "99", "153", "30", "1", "23", "181", "481", "277", "17", "1", "28", "303", "1239", "1451", "323", "1", "34", "479", "2811", "5572", "2946", "193", "1", "40", "726", "5805", "17607", "17343", "3806", "71", "1", "47", "1055", "11148", "48401", "77708", "36872", "3188", "1", "54", "1492", "20219", "120018", "288476", "243007", "54386", "1496" ]	Triangle read by rows: T(n,k) is the number of unlabeled connected multigraphs with n edges on k nodes and degree >= 3 at each node, loops allowed, n >= 2, 1 <= k <= floor(2*n/3).
A360863	[ "0", "1", "3", "5", "13", "36", "99", "301", "980", "3345", "12036", "45399", "178420", "729149" ]	Number of unlabeled connected multigraphs with n edges and degree >= 3 at each node, loops allowed.
A360864	[ "0", "3", "15", "111", "1076", "13870", "220520", "4185406", "92235118", "2314204852", "65129484278", "2032179006943", "69640160993587", "2600585852722150", "105127528809344785", "4574251821427917425", "213171992131468465801", "10593983324971249199532", "559293301762878627195807", "31259896932477899016109585", "1844062168535890557437809526" ]	Number of unlabeled connected multigraphs with circuit rank n and degree >= 3 at each node, loops allowed.
A360865	[ "0", "1", "3", "6", "16", "48", "130", "403", "1293", "4346", "15318", "56604", "217802", "873022" ]	Number of unlabeled multigraphs with n edges and degree >= 3 at each node, loops allowed.
A360866	[ "0", "0", "1", "0", "1", "0", "1", "1", "0", "1", "3", "2", "0", "1", "4", "7", "0", "1", "6", "19", "6", "0", "1", "8", "40", "37", "6", "0", "1", "10", "71", "135", "56", "0", "1", "12", "117", "366", "338", "35", "0", "1", "15", "184", "858", "1417", "494", "20", "0", "1", "17", "270", "1778", "4670", "3494", "492", "0", "1", "20", "387", "3413", "13125", "17355", "6047", "251" ]	Triangle read by rows: T(n,k) is the number of unlabeled connected loopless multigraphs with n edges on k nodes and degree >= 3 at each node, n >= 2, 1 <= k <= floor(2*n/3).
A360867	[ "0", "0", "1", "1", "2", "6", "12", "32", "92", "273", "869", "2989", "10722", "40599" ]	Number of unlabeled connected loopless multigraphs with n edges and degree >= 3 at each node.
A360868	[ "0", "1", "4", "23", "172", "1848", "25684" ]	Number of unlabeled connected loopless multigraphs with circuit rank n and degree >= 3 at each node.
A360869	[ "0", "0", "1", "1", "2", "7", "13", "35", "101", "295", "928", "3168", "11247", "42263" ]	Number of unlabeled loopless multigraphs with n edges and degree >= 3 at each node.
A360870	[ "1", "1", "2", "1", "4", "1", "7", "2", "1", "10", "8", "2", "1", "14", "19", "11", "1", "18", "40", "48", "7", "1", "23", "77", "154", "70", "5", "1", "28", "132", "421", "392", "71", "1", "34", "217", "1008", "1638", "690", "35", "1", "40", "340", "2210", "5623", "4548", "767", "16", "1", "47", "510", "4477", "16745", "22657", "8594", "566", "1", "54", "742", "8557", "44698", "92844", "64716", "11247", "226" ]	Triangle read by rows: T(n,k) is the number of unlabeled connected multigraphs with n edges on k nodes, no cut-points and degree >= 3 at each node, loops allowed, n >= 2, 1 <= k <= floor(2*n/3).
A360871	[ "0", "0", "2", "4", "9", "20", "44", "113", "329", "1044", "3622", "13544", "53596", "223084", "969158" ]	Number of unlabeled nonseparable (or 2-connected) multigraphs with n edges and degree >= 3 at each node, loops allowed.
A360873	[ "1", "3", "3", "7", "13", "7", "15", "51", "51", "15", "31", "205", "397", "205", "31", "63", "843", "3303", "3303", "843", "63", "127", "3493", "27877", "55933", "27877", "3493", "127", "255", "14451", "233751", "943095", "943095", "233751", "14451", "255", "511", "59485", "1938517", "15678925", "31450861", "15678925", "1938517", "59485", "511" ]	Array read by antidiagonals: T(m,n) is the number of (non-null) connected induced subgraphs in the rook graph K_m X K_n.
A360874	[ "3", "13", "51", "205", "843", "3493", "14451", "59485", "243483", "991573", "4021251", "16253965", "65530923", "263685253", "1059458451", "4252051645", "17050991163", "68332580533", "273716694051", "1096026940525", "4387590352203", "17560813373413", "70274617776051", "281192580728605", "1125052685342043" ]	Number of (non-null) connected induced subgraphs in the 2 X n rook graph.

A360768

[ "18", "24", "36", "48", "50", "54", "72", "75", "80", "90", "96", "98", "100", "108", "112", "120", "126", "135", "144", "147", "150", "160", "162", "168", "180", "189", "192", "196", "198", "200", "216", "224", "225", "234", "240", "242", "245", "250", "252", "264", "270", "288", "294", "300", "306", "312", "320", "324", "336", "338", "342", "350", "352", "360", "363", "375", "378", "384", "392", "396", "400", "405", "408" ]

Numbers k that are neither prime powers nor squarefree, such that k/rad(k) >= q, where rad(k) = A007947(k) and prime q = A119288(k).

A360769

[ "45", "63", "75", "99", "117", "135", "147", "153", "171", "175", "189", "207", "225", "245", "261", "275", "279", "297", "315", "325", "333", "351", "363", "369", "375", "387", "405", "423", "425", "441", "459", "475", "477", "495", "507", "513", "525", "531", "539", "549", "567", "575", "585", "603", "605", "621", "637", "639", "657", "675", "693", "711", "725", "735", "747", "765", "775", "783", "801", "819", "825" ]

Odd numbers that are neither prime powers nor squarefree.

A360770

[ "1", "5", "27", "260", "3125", "46684", "823543", "16777472", "387420498", "10000003125", "285311670611", "8916100495009", "302875106592253", "11112006826381559", "437893890380860625", "18446744073726328848", "827240261886336764177", "39346408075296925015353" ]

Expansion of Sum_{k>0} (x * (k + x^k))^k.

A360771

[ "1", "2", "5", "8", "20", "32", "77", "128", "288", "518", "1104", "2048", "4313", "8192", "16832", "32848", "66568", "131072", "264688", "524288", "1053737", "2097824", "4205568", "8388608", "16803744", "33554442", "67162112", "134222336", "268550704", "536870912", "1073999165", "2147483648", "4295493376", "8589962752" ]

Expansion of Sum_{k>=0} (x * (2 + x^k))^k.

A360772

[ "1", "2", "4", "5", "8", "9", "13", "16", "25", "32", "34", "64", "81", "89", "125", "128", "169", "233", "256", "441", "512", "610", "625", "729", "1024", "1156", "1597", "2048", "2197", "3025", "3125", "4096", "4181", "6561", "7921", "8192", "10946", "15625", "16384", "20736", "28561", "28657", "32768", "39304", "54289", "59049", "65536", "75025", "78125" ]

List of distinct numbers that are powers of odd-indexed Fibonacci numbers or even powers of nonzero even-indexed Fibonacci numbers.

A360773

[ "0", "1", "8", "1024", "620448" ]

Number of ways to tile a 2n X 2n square using rectangles with distinct dimensions such that the sum of the rectangles perimeters equals the area of the square.

A360774

[ "1", "1", "5", "31", "284", "3390", "49878", "871465", "17620450", "404554997", "10394845097", "295485704544", "9205957047661", "311922101632409", "11419004058232897", "449146827324857447", "18889836751306735360", "845892838094616177138", "40182354573647684880446" ]

Expansion of Sum_{k>=0} (x * (k + x))^k.

A360775

[ "1", "1", "4", "28", "260", "3152", "46913", "826677", "16823968", "388245283", "10016796672", "285699444297", "8926107792609", "303160590533808", "11120927427841820", "438196895219227683", "18457860168281435172", "827678295600605015006", "39364859979651634985089" ]

Expansion of Sum_{k>=0} (x * (k + x^2))^k.

A360776

[ "1", "1", "4", "27", "257", "3129", "46683", "823799", "16780342", "387467154", "10000823639", "285328449077", "8916487888186", "302885106945216", "11112292144568909", "437902806653498835", "18447046953316227905", "827251374022851280231", "39346845973273509115167" ]

Expansion of Sum_{k>=0} (x * (k + x^3))^k.

A360777

[ "9", "160", "143", "2679", "19933", "115248", "45", "1995" ]

a(n) is the index of the least n-gonal number that is the sum of two or more consecutive nonzero n-gonal numbers in more than one way, or -1 if no such number exists.

A360778

[ "1", "0", "0", "5", "4", "3", "2", "1", "0", "0", "2", "1", "0", "5", "4", "3", "2", "1", "0", "5", "4", "3", "2", "1", "0", "11", "10", "9", "8", "7", "6", "5", "4", "3", "2", "1", "0", "3", "2", "1", "0", "15", "14", "13", "12", "11", "10", "9", "8", "7", "6", "5", "4", "3", "2", "1", "0", "3", "2", "1", "0", "11", "10", "9", "8", "7", "6", "5", "4", "3", "2", "1", "0", "7", "6", "5", "4", "3", "2", "1", "0", "3", "2", "1", "0", "3", "2", "1", "0" ]

Smallest number k such that n + k is a refactorable number.

A360779

[ "1", "6", "1", "3", "6", "6", "12", "4", "16", "4", "12", "8", "4", "4", "8", "8", "4", "20", "4", "4", "16", "4", "24", "4", "20", "21", "3", "4", "8", "8", "4", "24", "12", "8", "32", "16", "4", "12", "12", "4", "8", "12", "28", "17", "3", "4", "2", "18", "4", "8", "8", "4", "12", "12", "20", "24", "4", "4", "16", "16", "12", "13", "7", "4", "4", "24", "8", "12", "24", "4", "8", "12", "44", "16", "12", "4", "16", "4", "24" ]

Refactorable numbers gaps: differences between consecutive refactorable numbers.

A360780

[ "1", "2", "8", "8", "8", "8", "8", "8", "9", "12", "12", "12", "18", "18", "18", "18", "18", "18", "24", "24", "24", "24", "24", "24", "36", "36", "36", "36", "36", "36", "36", "36", "36", "36", "36", "36", "40", "40", "40", "40", "56", "56", "56", "56", "56", "56", "56", "56", "56", "56", "56", "56", "56", "56", "56", "56", "60", "60", "60", "60", "72", "72", "72", "72", "72", "72", "72", "72", "72" ]

Least refactorable number > n.

A360781

[ "2", "3", "5", "7", "17", "19", "23", "29", "31", "37", "41", "43", "47", "53", "59", "61", "67", "71", "73", "79", "83", "101", "103", "107", "109", "113", "131", "139", "149", "151", "157", "163", "173", "179", "191", "193", "197", "211", "223", "227", "233", "239", "241", "251", "257", "263", "269", "271", "277", "281", "283", "293", "307", "311", "313", "317", "331" ]

Primes p such that at least one number remains prime when p is bracketed by a single digit d; that is, at least one instance of d//p//d is prime where // means concatenation.

A360782

[ "1", "1", "1", "3", "7", "16", "45", "125", "363", "1127", "3561", "11696", "39727", "138113", "494213", "1811075", "6784115", "25985928", "101520833", "404305549", "1640002039", "6767576175", "28395916893", "121048681024", "523902418555", "2300906314849", "10248029334297", "46266088140291" ]

Expansion of Sum_{k>=0} x^k / (1 - k*x^2)^(k+1).

A360783

[ "1", "1", "1", "1", "3", "7", "13", "24", "55", "133", "301", "678", "1639", "4120", "10253", "25591", "65869", "173551", "459493", "1225379", "3325123", "9162046", "25451181", "71296499", "202144225", "579612934", "1675822453", "4885178596", "14376297345", "42690792651", "127757371105", "385241085261", "1170960103855" ]

Expansion of Sum_{k>=0} x^k / (1 - k*x^3)^(k+1).

A360784

[ "1", "1", "3", "8", "18", "39", "86", "175", "352", "688", "1318", "2472", "4576", "8322", "14959", "26560", "46657", "81130", "139866", "239047", "405496", "682891", "1142466", "1899344", "3139432", "5160455", "8438871", "13732292", "22242647", "35867937", "57597730", "92121145", "146775205", "232998683", "368579188", "581091003" ]

Number of multisets of nonempty strict integer partitions with a total of n parts and total sum of 2n.

A360785

[ "1", "2", "5", "12", "26", "54", "112", "220", "427", "812", "1518", "2790", "5074", "9096", "16144", "28360", "49367", "85180", "145867", "247886", "418426", "701702", "1169673", "1938498", "3195497", "5240386", "8552308", "13892638", "22468406", "36184636", "58040397", "92737842", "147631545", "234184172", "370215442", "583343070" ]

Number of multisets of nonempty strict integer partitions with a total of 2n parts and total sum of 3n.

A360786

[ "0", "2", "42", "400", "2840", "17376", "97440", "516608", "2634624", "13058560", "63320576", "301707264", "1417009152", "6575120384", "30195425280", "137430827008", "620604391424", "2783097520128", "12403773407232", "54975376916480", "242441862512640", "1064326263734272", "4653131038195712", "20266193591992320" ]

Number of ways to place two dimers on an n-cube.

A360787

[ "1", "1", "1", "3", "13", "40", "177", "965", "4733", "28103", "184065", "1191888", "8713549", "67005689", "528870257", "4526024267", "40051790333", "368513578472", "3583302492545", "35868588067501", "373781214260749", "4052932682659599", "45218033687522481", "523234757502985824", "6245693941097387773" ]

Expansion of Sum_{k>=0} x^k / (1 - (k*x)^2)^(k+1).

A360788

[ "1", "1", "1", "1", "3", "25", "109", "324", "1135", "8803", "64189", "337854", "1707319", "13421410", "121248893", "894378619", "6082868725", "53046554917", "543432115477", "4989423130739", "42565774604131", "421544374075072", "4781440892689533", "51342685464272591", "522295380717090265" ]

Expansion of Sum_{k>=0} x^k / (1 - (k*x)^3)^(k+1).

A360789

[ "2", "3", "5", "7", "379", "23", "401", "61", "59", "29", "67", "71", "467", "79", "83", "179", "431", "89", "176557", "191", "24419", "491", "97", "101", "499", "1213", "3169", "3191", "523", "229", "3187", "223", "3203", "8609", "3163", "251", "176509", "257", "24509", "263", "3253", "269", "547", "3347", "1304867", "293" ]

Least prime p such that p mod primepi(p) = n.

A360790

[ "8", "13", "41", "53", "137", "173", "305", "397", "533", "877", "977", "1373", "1697", "1885", "2245", "2813", "3517", "3737", "4493", "5077", "5345", "6277", "6953", "7937", "9413", "10217", "10613", "11465", "12077", "12785", "16165", "17165", "18869", "19325", "22237", "22837", "24665", "26605", "27925", "29933", "32141", "32765", "36497", "37253", "38953", "39745" ]

Squared length of diagonal of right trapezoid with three consecutive prime length sides.

A360791

[ "1", "2", "4", "14", "28", "94", "218", "588", "1366", "3618", "8134", "20320", "45592", "105810", "236960", "539392", "1174530", "2612436", "5628606", "12226350", "26130568", "55938126", "117997774", "249680514", "523032956", "1094500962", "2275886514", "4727461792", "9762182762", "20148991512", "41403646304", "84961079990" ]

Sum of all prime encoded complete partitions of n.

A360792

[ "0", "0", "1", "2", "3", "4", "5", "7", "9", "10", "12", "15", "17", "19", "22", "25", "28", "31", "34", "37", "41", "45", "49", "53", "57", "61", "66", "71", "75", "80", "86", "91", "96", "102", "108", "114", "120", "126", "133", "139", "146", "153", "160", "167", "175", "182", "190", "198", "206", "214", "223", "231", "240", "249", "258", "267", "276", "285", "295", "305" ]

Integer portion of area of inscribed circle in a regular polygon having n sides of unit length.

A360793

[ "24", "40", "54", "56", "88", "104", "120", "135", "136", "152", "168", "184", "189", "232", "248", "250", "264", "270", "280", "296", "297", "312", "328", "344", "351", "375", "376", "378", "408", "424", "440", "456", "459", "472", "488", "513", "520", "536", "552", "568", "584", "594", "616", "621", "632", "664", "680", "686", "696", "702", "712", "728", "744", "750" ]

Numbers of the form m*p^3, where m > 1 is squarefree and prime p does not divide m.

A360794

[ "1", "3", "4", "11", "6", "43", "8", "109", "100", "281", "12", "1507", "14", "1863", "3376", "6937", "18", "26245", "20", "53211", "63022", "67739", "24", "572413", "78776", "372945", "1087048", "1761719", "30", "7362871", "32", "9947953", "16897486", "10027349", "8011116", "123101515", "38", "49807779", "241823440", "361722421", "42" ]

Expansion of Sum_{k>0} x^k / (1 - k * x^k)^(k+1).

A360795

[ "1", "3", "4", "17", "6", "211", "8", "1929", "7300", "22601", "12", "1724809", "14", "6703047", "223678576", "738787345", "18", "65630598229", "20", "2119646503661", "24448573943662", "3423809253371", "24", "21453113652593665", "12016296386718776", "4240253019018225", "8255251542208471048", "67251293544533119589", "30" ]

Expansion of Sum_{k>0} x^k / (1 - (k * x)^k)^(k+1).

A360796

[ "7", "9", "11", "13", "14", "17", "17", "19", "20", "25", "23", "29", "26", "27", "29", "37", "31", "40", "34", "35", "38", "46", "39", "41", "44", "43", "44", "54", "47", "58", "49", "51", "56", "53", "54", "67", "62", "59", "59", "70", "62", "73", "64", "65", "74", "78", "69", "71", "71", "75", "74", "86", "76", "77", "79", "83", "92", "93", "83", "103" ]

a(n) > n is the smallest integer such that there exist integers n < c <= d < a(n) satisfying n^2 + a(n)^2 = c^2 + d^2.

A360797

[ "1", "5", "13", "39", "81", "225", "449", "1115", "2345", "5373", "11265", "25483", "53249", "116497", "246405", "529195", "1114113", "2372741", "4980737", "10515511", "22025617", "46204953", "96468993", "201506607", "419432417", "872787997", "1811981789", "3758970975", "7784628225", "16108217801", "33285996545", "68723976779" ]

Expansion of Sum_{k>0} x^k / (1 - 2 * x^k)^(k+1).

A360798

[ "1", "5", "13", "45", "81", "321", "449", "1745", "2945", "9153", "11265", "60609", "53249", "230401", "410625", "1259777", "1114113", "7263233", "4980737", "31337473", "44630017", "115367937", "96468993", "937283585", "551550977", "2399256577", "4594597889", "14579646465", "7784628225", "89894944769", "33285996545" ]

Expansion of Sum_{k>0} x^k / (1 - (2 * x)^k)^(k+1).

A360801

[ "1", "3", "5", "13", "17", "51", "65", "169", "281", "603", "1025", "2373", "4097", "8655", "16685", "33969", "65537", "134151", "262145", "530269", "1050481", "2108439", "4194305", "8420201", "16778337", "33607707", "67120565", "134338493", "268435457", "537151131", "1073741825", "2148024289", "4295035145", "8591048739" ]

Expansion of Sum_{k>0} (x / (1 - 2 * x^k))^k.

A360802

[ "1", "3", "5", "17", "17", "105", "65", "449", "641", "1953", "1025", "16257", "4097", "37761", "93185", "247809", "65537", "1499649", "262145", "6596609", "8847361", "13654017", "4194305", "210026497", "90177537", "251764737", "833880065", "2659418113", "268435457", "18345328641", "1073741825", "53553922049", "75438751745" ]

Expansion of Sum_{k>0} (x / (1 - (2 * x)^k))^k.

A360803

[ "3", "313", "94863", "298327", "987917", "3162083", "9893887", "29983327", "99477133", "99483667", "197483417", "282753937", "314623583", "315432874", "706399164", "773303937", "894303633", "947047833", "948675387", "989938887", "994927133", "994987437", "998398167", "2428989417", "2754991833", "2983284917", "2999833327" ]

Numbers whose squares have a digit average of 8 or more.

A360804

[ "1", "1", "21", "253", "2401", "36237", "815929", "18713197" ]

Number of ways to tile an n X n square using rectangles with distinct areas.

A360805

[ "0", "31", "120", "283", "293", "712", "2872", "3287", "5028", "5129", "7088", "9553", "13229", "14232", "14799", "15113", "20153", "20830", "23239", "30233", "31430", "31667", "34443", "40654", "44298", "50184", "78877", "105834", "115281", "125120", "164253", "192103", "201590", "227747", "239910", "241910", "282230", "322550", "374370" ]

Nonnegative integers k such that k! mod nextprime(k) is larger than k.

A360806

[ "1", "2", "8", "80", "2240", "215040", "77414400", "61931520000", "170930995200000", "1340099002368000000" ]

a(0) = 1; for n >= 1, a(n) is the least integer k > a(n-1) such that k / A000005(k) = a(n-1).

A360808

[ "1", "2", "2", "2", "8", "8", "35", "16", "51", "145", "1112", "1145", "10929", "41400", "542785", "40384", "583169", "2781808", "48558706", "65461347", "1277941540", "7370563251", "159694747220", "63387056365", "1500631724572", "10152855622657" ]

Number of double cosets of the Sylow 2-subgroup of the symmetric group S_n.

A360809

[ "2", "5", "8", "6", "7", "0", "5", "0", "5", "9", "7", "8", "6", "8", "0", "8", "2", "2", "7", "7", "7", "8", "1", "0", "6", "8", "7", "2", "9", "4", "6", "9", "6", "0", "2", "1", "3", "5", "7", "3", "0", "9", "6", "2", "7", "4", "2", "4", "8", "9", "3", "6", "1", "2", "4", "4", "6", "7", "0", "8", "2", "4", "2", "2", "5", "8", "5", "9", "4", "0", "4", "5", "5", "6", "0", "6", "6", "4", "3", "4", "2", "6", "4", "2", "8", "8", "2", "7", "7", "7", "5", "6", "7", "5", "3", "9", "0", "8", "8", "7", "6", "4", "4", "6", "9", "9", "8", "1" ]

Decimal expansion of the area under the curve of the reciprocal of the Luschny factorial function from zero to infinity.

A360810

[ "1", "1", "1", "2", "5", "11", "29", "81", "229", "696", "2181", "7045", "23653", "81433", "288173", "1046814", "3887749", "14768783", "57275541", "226462801", "912443397", "3741515804", "15603500797", "66134448329", "284660214181", "1243605590897", "5511058189989", "24760003963802", "112726590916645" ]

Expansion of Sum_{k>=0} ( x / (1 - k * x^2) )^k.

A360811

[ "1", "1", "1", "1", "2", "5", "10", "18", "38", "91", "211", "472", "1108", "2754", "6881", "17101", "43443", "113565", "300142", "797191", "2147414", "5883976", "16293712", "45471429", "128285353", "366266188", "1055534118", "3066483484", "8989837397", "26602652605", "79370560477", "238606427241", "722973445270" ]

Expansion of Sum_{k>=0} ( x / (1 - k * x^3) )^k.

A360812

[ "1", "1", "1", "2", "9", "29", "113", "613", "3033", "17010", "110929", "713249", "5061097", "38762873", "302389553", "2544613578", "22404995001", "203762678941", "1960880744337", "19509713674397", "201306862742217", "2166901479447194", "24018963506471921", "275731857268608673", "3271769647891351705" ]

Expansion of Sum_{k>=0} ( x / (1 - (k * x)^2) )^k.

A360813

[ "1", "1", "1", "1", "2", "17", "82", "258", "818", "5671", "43363", "240520", "1183168", "8547054", "77831681", "596258173", "4031934111", "33313129161", "338733239446", "3187239159511", "27197807726066", "260179611473044", "2918973182685904", "31820249821418229", "324099587971865989" ]

Expansion of Sum_{k>=0} ( x / (1 - (k * x)^3) )^k.

A360814

[ "1", "0", "1", "2", "4", "10", "30", "98", "338", "1240", "4877", "20496", "91213", "426678", "2090081", "10702438", "57193760", "318283388", "1840036058", "11026424446", "68370955450", "438039068726", "2896018310881", "19733372875632", "138418266287689", "998363508783924", "7396739279819185", "56239695790595786" ]

Expansion of Sum_{k>=0} x^(2*k) / (1 - k*x)^(k+1).

A360815

[ "1", "0", "0", "1", "2", "3", "5", "11", "30", "88", "260", "771", "2343", "7474", "25380", "91650", "347988", "1371873", "5570173", "23233703", "99676434", "440931977", "2014619700", "9506385864", "46246356169", "231348803925", "1187212953132", "6239006165820", "33546182775824", "184497923546700" ]

Expansion of Sum_{k>=0} x^(3*k) / (1 - k*x)^(k+1).

A360816

[ "1", "0", "1", "2", "19", "100", "1118", "10034", "134993", "1715140", "27589661", "449763360", "8522965956", "168431719308", "3698624353289", "85523954588806", "2142927489388319", "56618555339223572", "1596938935380604858", "47399670488829289678", "1487559109670284821841" ]

Expansion of Sum_{k>=0} (k*x)^(2*k) / (1 - k*x)^(k+1).

A360817

[ "1", "0", "0", "1", "2", "3", "68", "389", "1542", "24810", "251564", "1814487", "27520734", "391640548", "4295115396", "69305652406", "1221344986380", "18207710383335", "329699350020676", "6759819628538561", "126950556666301050", "2624697847966227077", "60825028694289947940", "1365568620213461601924" ]

Expansion of Sum_{k>=0} (k*x)^(3*k) / (1 - k*x)^(k+1).

A360818

[ "1", "0", "1", "1", "17", "65", "922", "7074", "106183", "1248479", "21144289", "331763177", "6441011484", "124904347404", "2773880604749", "63538143151589", "1600211849569585", "42076439530000297", "1189408501356380558", "35214128238218917974", "1106088535644470694779" ]

Expansion of Sum_{k>=0} ( (k*x)^2 / (1 - k*x) )^k.

A360819

[ "1", "0", "0", "1", "1", "1", "65", "257", "769", "21732", "182268", "1075171", "22120299", "292415838", "2784944366", "52394511682", "914813711338", "12411977351379", "240868108545883", "5024364548461861", "88977315031536205", "1888119425325238979", "44744897995532996819", "971263427084750362992" ]

Expansion of Sum_{k>=0} ( (k*x)^3 / (1 - k*x) )^k.

A360820

[ "1", "4", "48", "1792", "221184", "98566144", "173946175488", "1281755680079872", "39534286378918477824", "5018464395368794081460224", "2586745980900067184722499862528", "5375203895735606878055792019528220672", "44865714160227204455469409035569750630989824", "1501355804811017489524770237231795462175548447391744" ]

a(n) = Sum_{k=0..n} binomial(n, k)*2^(n^2-k*(n-k)).

A360821

[ "0", "1", "1", "1", "1", "1", "0", "1", "1", "2", "1", "2", "2", "2", "2", "2", "2", "2", "3", "3", "2", "2", "2", "2", "1", "2", "1", "1", "2", "2", "2", "2", "2", "2", "2", "2", "1", "1", "2", "3", "2", "2", "2", "2", "2", "2", "3", "3", "3", "3", "3", "3", "4", "4", "3", "3", "2", "2", "2", "3", "3", "3", "3", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "4", "3", "3", "3", "4", "4", "4", "4", "4", "5", "5", "5", "5", "5", "6" ]

Number of primes of the form k^2+1 between n^2 and 2*n^2 exclusive.

A360822

[ "1", "2", "3", "4", "5", "6", "7", "8", "9", "13", "14", "17", "22", "23", "27", "28", "29", "30", "31", "33", "43", "53", "63", "67", "77", "83", "91", "93", "94", "97", "99", "141", "167", "173", "197", "283", "293", "297", "298", "303", "313", "314", "316", "447", "583", "707", "767", "833", "836", "917", "943", "947", "1378", "2917", "2983", "3033", "5467", "9417", "9433", "29983", "31367", "94863" ]

Numbers whose squares have at most 2 digits less than 8.

A360823

[ "1", "4", "6", "20", "10", "96", "14", "256", "288", "650", "22", "4200", "26", "4004", "11160", "18784", "34", "70758", "38", "164140", "196098", "136664", "46", "1756728", "393800", "747890", "3287844", "5452076", "58", "22563060", "62", "31220032", "50767926", "20059286", "41640130", "391194396", "74", "99622016", "725647728", "1298396440" ]

Expansion of Sum_{k>0} k * x^k / (1 - k * x^k)^(k+1).

A360824

[ "1", "6", "30", "284", "3130", "47082", "823550", "16782664", "387422928", "10000094720", "285311670622", "8916102486528", "302875106592266", "11112006871683606", "437893890382576560", "18446744074918103056", "827240261886336764194", "39346408075331452862196" ]

Expansion of Sum_{k>0} (k * x)^k / (1 - k * x^k)^(k+1).

A360825

[ "1", "1", "2", "2", "4", "1", "6", "2", "5", "1", "10", "1", "12", "3", "8", "1", "16", "1", "18", "4", "11", "1", "22", "22", "6", "5", "14", "1", "28", "1", "30", "33", "20", "31", "18", "1", "36", "7", "20", "1", "40", "1", "42", "8", "23", "1", "46", "19", "11", "9", "26", "1", "52", "30", "27", "10", "29", "1", "58", "1", "60", "43", "53", "56", "33", "1", "66", "12", "35", "1", "70", "1", "72", "27", "23" ]

a(n) is the remainder after dividing n! by its least nondivisor.

A360827

[ "443", "647", "1847", "2243", "2687", "2699", "6263", "6563", "7487", "7583", "8627", "8663", "9419", "9767", "10223", "11867", "12323", "13187", "13907", "14627", "14723", "14783", "17747", "17783", "19739", "20639", "20807", "21863", "22307", "23747", "24107", "24923", "25127", "26759", "27983", "29207", "29819", "30839", "31247", "32303", "34403", "34439" ]

Primes p, not safe primes, such that the smallest factor of (2^(p-1)-1) / 3 is equal to p.

A360828

[ "1", "6", "6", "4", "8", "2", "2", "8", "4", "2", "9", "7", "8", "3", "3", "9", "3", "9", "4", "0", "0", "3", "0", "9", "5", "7", "2", "8", "2", "9", "0", "6", "5", "6", "9", "8", "1", "7", "4", "3", "0", "2", "2", "8", "5", "8", "6", "1", "4", "0", "9", "9", "6", "8", "9", "6", "4", "7", "1", "0", "8", "3", "2", "2", "7", "3", "6", "5", "6", "3", "9", "4", "5", "6", "3", "5", "4", "8", "6", "3", "2", "3", "6", "3", "0", "9", "2", "7", "3", "3", "4", "6", "1", "8", "3", "7", "2", "2", "9", "4" ]

Decimal expansion of the ratio between the perimeter of the first Morley triangle of an isosceles right triangle and the perimeter of this isosceles right triangle.

A360829

[ "3", "1", "0", "8", "8", "9", "1", "3", "2", "4", "5", "5", "3", "5", "2", "6", "3", "6", "7", "3", "0", "3", "1", "0", "9", "7", "6", "3", "5", "2", "7", "6", "6", "4", "2", "1", "4", "9", "9", "0", "9", "1", "9", "4", "1", "6", "8", "1", "6", "6", "0", "9", "9", "0", "9", "7", "6", "6", "2", "2", "1", "4", "0", "4", "0", "8", "8", "2", "7", "7", "9", "5", "9", "0", "4", "0", "0", "0", "6", "4", "8", "9", "2", "0", "0", "5", "8", "2", "6", "8", "2", "5", "1", "8", "5", "0", "0", "8" ]

Decimal expansion of the ratio between the area of the first Morley triangle of an isosceles right triangle and its area.

A360830

[ "1", "3", "6", "42", "84", "252", "2772", "36036", "612612", "11639628", "267711444", "803134332", "23290895628", "722017764468", "1444035528936", "53429314570632", "2190601897395912", "94195881588024216", "4427206434637138152", "30990445042459967064" ]

Numbers that when concatenated with the natural numbers from 1 to N are divisible by the corresponding order number.

A360831

[ "1", "6", "30", "308", "3130", "49962", "823550", "17107464", "387617328", "10058609120", "285311670622", "8931600297696", "302875106592266", "11117432610599574", "437894531752211760", "18449277498826162192", "827240261886336764194", "39347911865350001626164" ]

Expansion of Sum_{k>0} (k * x)^k / (1 - (k * x)^k)^(k+1).

A360832

[ "1", "1", "4", "28", "288", "3855", "63232", "1227291", "27511296", "699389444", "19880700928", "624817997477", "21512488648704", "805233062024021", "32556682898653184", "1413981749074790444", "65652661019642560512", "3245240681196968168619", "170146759140135777861632" ]

Expansion of Sum_{k>=0} ( k * x / (1 - (k * x)^2) )^k.

A360833

[ "1", "1", "4", "27", "257", "3189", "48843", "889080", "18731109", "448004763", "11987812504", "354763577414", "11503684020051", "405589341060930", "15447798292502206", "632069580794524857", "27649951709582591394", "1287748889361331630661", "63616184683123273364961" ]

Expansion of Sum_{k>=0} ( k * x / (1 - (k * x)^3) )^k.

A360834

[ "1", "1", "4", "29", "304", "4100", "67520", "1314167", "29520128", "751658635", "21393444864", "673046604600", "23192501108736", "868730852002205", "35145114836811776", "1527192185786650417", "70941146068492943360", "3508043437942077557884", "183989995827118805352448" ]

Expansion of Sum_{k>=0} (k * x)^k / (1 - (k * x)^2)^(k+1).

A360835

[ "1", "1", "4", "27", "258", "3221", "49572", "905466", "19122502", "458161191", "12275530636", "363646493044", "11801356347294", "416365459777150", "15867258718677348", "649548679156603983", "28426564854590132236", "1324406974148881529057", "65448443631801436742052" ]

Expansion of Sum_{k>=0} (k * x)^k / (1 - (k * x)^3)^(k+1).

A360836

[ "12880", "18896570" ]

a(n) is the least n-gonal pyramidal number that is the sum of two or more consecutive nonzero n-gonal pyramidal numbers in more than one way.

A360837

[ "1", "3", "59", "10079", "744666", "163710521" ]

a(n) is the least positive integer that can be expressed as the sum of one or more consecutive prime-indexed primes in exactly n ways.

A360839

[ "1", "6", "32", "103", "250", "220" ]

Number of minimal graphs of twin-width 2 on n unlabeled vertices.

A360840

[ "432", "2592", "139968", "444528", "472392", "995328", "3456000", "5174928", "6561000", "10125000", "15552000", "15804072", "17496000", "25299648", "28449792", "37340352", "54675000", "63700992", "85957848", "88723728", "99574272", "120891312", "144027072", "169869312", "177147000", "197413632", "253125000", "259308000" ]

3-full numbers (A036966) sandwiched between twin primes.

A360841

[ "2592", "139968", "995328", "37340352", "63700992", "99574272", "169869312", "414720000", "1399680000", "4076863488", "10871635968", "17714700000", "22781250000", "25312500000", "35888419872", "51840000000", "82012500000", "98802571392", "135304020000", "136136700000", "170749797552", "174960000000", "196730062848" ]

4-full numbers (A036967) sandwiched between twin primes.

A360842

[ "139968", "995328", "63700992", "4076863488", "17714700000", "82012500000", "98802571392", "174960000000", "445240556352", "641194278912", "889223142528", "1059917571072", "1594323000000", "1663012435968", "2348273369088", "3333709317312", "5717741400000", "16260080320512", "19144761127488", "28697814000000" ]

5-full numbers (A069492) sandwiched between twin primes.

A360843

[ "139968", "98802571392", "174960000000", "889223142528", "1594323000000", "2348273369088", "19144761127488", "28697814000000", "56358560858112", "84537841287168", "150289495621632", "186624000000000", "328341017826432", "369056250000000", "392147405854848", "578415690713088", "597871125000000" ]

6-full numbers (A069493) sandwiched between twin primes.

A360844

[ "4", "432", "2592", "139968", "139968", "174960000000", "56358560858112", "84537841287168", "578415690713088", "578415690713088", "1141260857376768", "61628086298345472", "61628086298345472", "61628086298345472", "322850407500000000000000000000", "322850407500000000000000000000", "62518864539857068333550694039552" ]

a(n) is the least k-full number that is sandwiched between twin primes.

A360846

[ "1", "3", "3", "4", "8", "4", "4", "17", "17", "4", "4", "32", "65", "32", "4", "4", "66", "222", "222", "66", "4", "4", "130", "766", "1280", "766", "130", "4", "4", "262", "2685", "7629", "7629", "2685", "262", "4", "4", "522", "9450", "46032", "78981", "46032", "9450", "522", "4", "4", "1046", "33158", "278419", "820308", "820308", "278419", "33158", "1046", "4" ]

Array read by antidiagonals: T(m,n) is the number of dominating induced trees in the grid graph P_m X P_n.

A360847

[ "1", "8", "65", "1280", "78981", "14605388", "7904828158", "12456744197696", "57118103869618858", "760896261783236975004", "29416443122724544970455433", "3297715940113139272931793598648", "1071333966021766251746119497973623975", "1008129126269380724757869194465038817386728" ]

Number of dominating induced trees in the n X n grid graph.

A360848

[ "3", "8", "17", "32", "66", "130", "262", "522", "1046", "2090", "4182", "8362", "16726", "33450", "66902", "133802", "267606", "535210", "1070422", "2140842", "4281686", "8563370", "17126742", "34253482", "68506966", "137013930", "274027862", "548055722", "1096111446", "2192222890", "4384445782", "8768891562" ]

Number of dominating induced trees in the n-ladder graph P_2_X P_n.

A360849

[ "0", "0", "0", "0", "1", "0", "0", "3", "3", "0", "0", "6", "15", "6", "0", "0", "10", "42", "42", "10", "0", "0", "15", "90", "204", "90", "15", "0", "0", "21", "165", "660", "660", "165", "21", "0", "0", "28", "273", "1650", "3940", "1650", "273", "28", "0", "0", "36", "420", "3486", "15390", "15390", "3486", "420", "36", "0", "0", "45", "612", "6552", "45150", "113865", "45150", "6552", "612", "45", "0" ]

Array read by antidiagonals: T(m,n) is the number of (undirected) cycles in the complete bipartite graph K_{m,n}.

A360850

[ "1", "3", "3", "6", "12", "6", "10", "33", "33", "10", "15", "72", "135", "72", "15", "21", "135", "438", "438", "135", "21", "28", "228", "1140", "2224", "1140", "228", "28", "36", "357", "2511", "8850", "8850", "2511", "357", "36", "45", "528", "4893", "27480", "55725", "27480", "4893", "528", "45", "55", "747", "8700", "70462", "265665", "265665", "70462", "8700", "747", "55" ]

Array read by antidiagonals: T(m,n) is the number of (undirected) paths in the complete bipartite graph K_{m,n}.

A360851

[ "0", "1", "1", "3", "8", "3", "6", "27", "27", "6", "10", "64", "126", "64", "10", "15", "125", "426", "426", "125", "15", "21", "216", "1125", "2208", "1125", "216", "21", "28", "343", "2493", "8830", "8830", "2493", "343", "28", "36", "512", "4872", "27456", "55700", "27456", "4872", "512", "36", "45", "729", "8676", "70434", "265635", "265635", "70434", "8676", "729", "45" ]

Array read by antidiagonals: T(m,n) is the number of induced paths in the rook graph K_m X K_n.

A360852

[ "0", "8", "126", "2208", "55700", "2006280", "98309778", "6291829376", "509638185288", "50963818537800", "6166622043087110", "887993574204562848", "150070914040571147676", "29413899151951944980168", "6618127309189187620585050", "1694240591152432030869834240", "489635530843052856921382173968" ]

Number of induced paths in the n X n rook graph.

A360853

[ "0", "0", "0", "1", "1", "1", "4", "5", "5", "4", "10", "14", "21", "14", "10", "20", "30", "58", "58", "30", "20", "35", "55", "125", "236", "125", "55", "35", "56", "91", "231", "720", "720", "231", "91", "56", "84", "140", "385", "1754", "4040", "1754", "385", "140", "84", "120", "204", "596", "3654", "15550", "15550", "3654", "596", "204", "120" ]

Array read by antidiagonals: T(m,n) is the number of induced cycles in the rook graph K_m X K_n.

A360854

[ "0", "1", "21", "236", "4040", "114105", "4662721", "256485936", "18226110456", "1623855703785", "177195820502965", "23237493232958796", "3605437233380103056", "653193551573628910481", "136634950180317224879985", "32681589590709963123110080", "8863149183726257535369656976" ]

Number of induced cycles in the n X n rook graph.

A360855

[ "0", "0", "0", "1", "0", "1", "4", "2", "2", "4", "10", "8", "6", "8", "10", "20", "20", "16", "16", "20", "20", "35", "40", "35", "32", "35", "40", "35", "56", "70", "66", "60", "60", "66", "70", "56", "84", "112", "112", "104", "100", "104", "112", "112", "84", "120", "168", "176", "168", "160", "160", "168", "176", "168", "120", "165", "240", "261", "256", "245", "240", "245", "256", "261", "240", "165" ]

Array read by antidiagonals: T(m,n) is the number of triangles in the rook graph K_m X K_n.

A360856

[ "1", "1", "2", "6", "16", "48", "140", "424", "1280", "3920", "12032", "37184", "115248", "358624", "1118784", "3499584", "10969344", "34450944", "108377984", "341465344", "1077300224", "3403006464", "10761447424", "34065967104", "107937899264", "342293526016", "1086339120128", "3450236511232", "10965437349888" ]

a(n) = [x^n](1/2)*(1 + (2*x + 1)/sqrt(1 - 8*x^2*(x + 1))).

A360857

[ "1", "1", "1", "1", "2", "6", "1", "3", "12", "12", "1", "4", "20", "30", "60", "1", "5", "30", "60", "150", "150", "1", "6", "42", "105", "315", "420", "700", "1", "7", "56", "168", "588", "980", "1960", "1960", "1", "8", "72", "252", "1008", "2016", "4704", "5880", "8820", "1", "9", "90", "360", "1620", "3780", "10080", "15120", "26460", "26460" ]

Triangle read by rows. T(n, k) = binomial(n, ceil(k/2)) * binomial(n + 1, floor(k/2)).

A360858

[ "1", "1", "2", "1", "3", "6", "1", "4", "12", "18", "1", "5", "20", "40", "60", "1", "6", "30", "75", "150", "200", "1", "7", "42", "126", "315", "525", "700", "1", "8", "56", "196", "588", "1176", "1960", "2450", "1", "9", "72", "288", "1008", "2352", "4704", "7056", "8820", "1", "10", "90", "405", "1620", "4320", "10080", "17640", "26460", "31752" ]

Triangle read by rows. T(n, k) = binomial(n + 1, ceil(k/2)) * binomial(n, floor(k/2)).

A360859

[ "1", "1", "1", "1", "2", "4", "1", "3", "9", "9", "1", "4", "16", "24", "36", "1", "5", "25", "50", "100", "100", "1", "6", "36", "90", "225", "300", "400", "1", "7", "49", "147", "441", "735", "1225", "1225", "1", "8", "64", "224", "784", "1568", "3136", "3920", "4900", "1", "9", "81", "324", "1296", "3024", "7056", "10584", "15876", "15876", "1", "10", "100", "450", "2025", "5400", "14400", "25200", "44100", "52920", "63504" ]

Triangle read by rows. T(n, k) = binomial(n, ceil(k/2)) * binomial(n, floor(k/2)).

A360860

[ "1", "0", "1", "0", "1", "2", "0", "2", "4", "8", "0", "8", "14", "26", "64", "0", "64", "96", "144", "296", "1024", "0", "1024", "1344", "1664", "2424", "6064", "32768", "0", "32768", "38912", "42752", "48832", "70672", "230896", "2097152", "0", "2097152", "2326528", "2412544", "2497664", "2701504", "3823072", "16886864", "268435456" ]

Accumulation triangle of A360603 read by rows.

A360861

[ "1", "2", "7", "22", "81", "281", "1058", "3830", "14605", "54127", "208110", "782761", "3027038", "11501478", "44668692", "170974710", "666220005", "2564271875", "10018268150", "38728479647", "151631858378", "588229029258", "2307174835212", "8975958379817", "35258881445606", "137501193282026", "540821096592028" ]

a(n) = Sum_{k=0..n} binomial(n, ceil(k/2)) * binomial(n, floor(k/2)).

A360862

[ "1", "1", "2", "1", "4", "1", "7", "5", "1", "10", "20", "5", "1", "14", "48", "36", "1", "18", "99", "153", "30", "1", "23", "181", "481", "277", "17", "1", "28", "303", "1239", "1451", "323", "1", "34", "479", "2811", "5572", "2946", "193", "1", "40", "726", "5805", "17607", "17343", "3806", "71", "1", "47", "1055", "11148", "48401", "77708", "36872", "3188", "1", "54", "1492", "20219", "120018", "288476", "243007", "54386", "1496" ]

Triangle read by rows: T(n,k) is the number of unlabeled connected multigraphs with n edges on k nodes and degree >= 3 at each node, loops allowed, n >= 2, 1 <= k <= floor(2*n/3).

A360863

[ "0", "1", "3", "5", "13", "36", "99", "301", "980", "3345", "12036", "45399", "178420", "729149" ]

Number of unlabeled connected multigraphs with n edges and degree >= 3 at each node, loops allowed.

A360864

[ "0", "3", "15", "111", "1076", "13870", "220520", "4185406", "92235118", "2314204852", "65129484278", "2032179006943", "69640160993587", "2600585852722150", "105127528809344785", "4574251821427917425", "213171992131468465801", "10593983324971249199532", "559293301762878627195807", "31259896932477899016109585", "1844062168535890557437809526" ]

Number of unlabeled connected multigraphs with circuit rank n and degree >= 3 at each node, loops allowed.

A360865

[ "0", "1", "3", "6", "16", "48", "130", "403", "1293", "4346", "15318", "56604", "217802", "873022" ]

Number of unlabeled multigraphs with n edges and degree >= 3 at each node, loops allowed.

A360866

[ "0", "0", "1", "0", "1", "0", "1", "1", "0", "1", "3", "2", "0", "1", "4", "7", "0", "1", "6", "19", "6", "0", "1", "8", "40", "37", "6", "0", "1", "10", "71", "135", "56", "0", "1", "12", "117", "366", "338", "35", "0", "1", "15", "184", "858", "1417", "494", "20", "0", "1", "17", "270", "1778", "4670", "3494", "492", "0", "1", "20", "387", "3413", "13125", "17355", "6047", "251" ]

Triangle read by rows: T(n,k) is the number of unlabeled connected loopless multigraphs with n edges on k nodes and degree >= 3 at each node, n >= 2, 1 <= k <= floor(2*n/3).

A360867

[ "0", "0", "1", "1", "2", "6", "12", "32", "92", "273", "869", "2989", "10722", "40599" ]

Number of unlabeled connected loopless multigraphs with n edges and degree >= 3 at each node.

A360868

[ "0", "1", "4", "23", "172", "1848", "25684" ]

Number of unlabeled connected loopless multigraphs with circuit rank n and degree >= 3 at each node.

A360869

[ "0", "0", "1", "1", "2", "7", "13", "35", "101", "295", "928", "3168", "11247", "42263" ]

Number of unlabeled loopless multigraphs with n edges and degree >= 3 at each node.

A360870

[ "1", "1", "2", "1", "4", "1", "7", "2", "1", "10", "8", "2", "1", "14", "19", "11", "1", "18", "40", "48", "7", "1", "23", "77", "154", "70", "5", "1", "28", "132", "421", "392", "71", "1", "34", "217", "1008", "1638", "690", "35", "1", "40", "340", "2210", "5623", "4548", "767", "16", "1", "47", "510", "4477", "16745", "22657", "8594", "566", "1", "54", "742", "8557", "44698", "92844", "64716", "11247", "226" ]

Triangle read by rows: T(n,k) is the number of unlabeled connected multigraphs with n edges on k nodes, no cut-points and degree >= 3 at each node, loops allowed, n >= 2, 1 <= k <= floor(2*n/3).

A360871

[ "0", "0", "2", "4", "9", "20", "44", "113", "329", "1044", "3622", "13544", "53596", "223084", "969158" ]

Number of unlabeled nonseparable (or 2-connected) multigraphs with n edges and degree >= 3 at each node, loops allowed.

A360873

[ "1", "3", "3", "7", "13", "7", "15", "51", "51", "15", "31", "205", "397", "205", "31", "63", "843", "3303", "3303", "843", "63", "127", "3493", "27877", "55933", "27877", "3493", "127", "255", "14451", "233751", "943095", "943095", "233751", "14451", "255", "511", "59485", "1938517", "15678925", "31450861", "15678925", "1938517", "59485", "511" ]

Array read by antidiagonals: T(m,n) is the number of (non-null) connected induced subgraphs in the rook graph K_m X K_n.

A360874

[ "3", "13", "51", "205", "843", "3493", "14451", "59485", "243483", "991573", "4021251", "16253965", "65530923", "263685253", "1059458451", "4252051645", "17050991163", "68332580533", "273716694051", "1096026940525", "4387590352203", "17560813373413", "70274617776051", "281192580728605", "1125052685342043" ]

Number of (non-null) connected induced subgraphs in the 2 X n rook graph.