a-number
stringlengths 7
7
| sequence
sequencelengths 1
377
| description
stringlengths 3
852
|
|---|---|---|
A001101 | [
"18",
"21",
"27",
"42",
"45",
"63",
"84",
"111",
"114",
"117",
"133",
"152",
"153",
"156",
"171",
"190",
"195",
"198",
"201",
"207",
"209",
"222",
"228",
"247",
"261",
"266",
"285",
"333",
"370",
"372",
"399",
"402",
"407",
"423",
"444",
"465",
"481",
"511",
"516",
"518",
"531",
"555",
"558",
"592",
"603"
] | Moran numbers: k such that k/(sum of digits of k) is prime. |
A001102 | [
"1",
"2",
"3",
"4",
"5",
"6",
"7",
"8",
"9",
"12",
"24",
"36",
"48",
"81",
"100",
"144",
"150",
"192",
"200",
"225",
"288",
"300",
"320",
"324",
"375",
"400",
"441",
"500",
"512",
"600",
"640",
"648",
"700",
"704",
"735",
"800",
"832",
"882",
"900",
"960",
"1014",
"1088",
"1200",
"1452",
"1458",
"1521",
"1815",
"2023"
] | Numbers k such that k / (sum of digits of k) is a square. |
A001103 | [
"1",
"2",
"3",
"4",
"5",
"6",
"7",
"8",
"9",
"11",
"15",
"24",
"36",
"115",
"175",
"212",
"624",
"735",
"816",
"1115",
"1184",
"1197",
"1416",
"2144",
"3171",
"3276",
"3915",
"6624",
"7119",
"8832",
"9612",
"11133",
"11212",
"11331",
"12128",
"12216",
"12768",
"13131",
"21184",
"21728",
"24912",
"31113",
"31488",
"32172",
"32616",
"35175"
] | Numbers n such that (n / product of digits of n) is 1 or a prime. |
A001104 | [
"1",
"2",
"3",
"4",
"5",
"6",
"7",
"8",
"9",
"135",
"144",
"384",
"1575",
"1715",
"6144",
"6912",
"11664",
"14112",
"16224",
"18816",
"23328",
"26136",
"31212",
"41616",
"82944",
"83232",
"93312",
"131424",
"131712",
"186624",
"248832",
"371112",
"1168128",
"2214144",
"2239488",
"2333772",
"3321216",
"3881472",
"6642432"
] | Numbers n such that n / product of digits of n is a square. |
A001105 | [
"0",
"2",
"8",
"18",
"32",
"50",
"72",
"98",
"128",
"162",
"200",
"242",
"288",
"338",
"392",
"450",
"512",
"578",
"648",
"722",
"800",
"882",
"968",
"1058",
"1152",
"1250",
"1352",
"1458",
"1568",
"1682",
"1800",
"1922",
"2048",
"2178",
"2312",
"2450",
"2592",
"2738",
"2888",
"3042",
"3200",
"3362",
"3528",
"3698",
"3872",
"4050",
"4232",
"4418"
] | a(n) = 2*n^2. |
A001106 | [
"0",
"1",
"9",
"24",
"46",
"75",
"111",
"154",
"204",
"261",
"325",
"396",
"474",
"559",
"651",
"750",
"856",
"969",
"1089",
"1216",
"1350",
"1491",
"1639",
"1794",
"1956",
"2125",
"2301",
"2484",
"2674",
"2871",
"3075",
"3286",
"3504",
"3729",
"3961",
"4200",
"4446",
"4699",
"4959",
"5226",
"5500",
"5781",
"6069",
"6364"
] | 9-gonal (or enneagonal or nonagonal) numbers: a(n) = n*(7*n-5)/2. |
A001107 | [
"0",
"1",
"10",
"27",
"52",
"85",
"126",
"175",
"232",
"297",
"370",
"451",
"540",
"637",
"742",
"855",
"976",
"1105",
"1242",
"1387",
"1540",
"1701",
"1870",
"2047",
"2232",
"2425",
"2626",
"2835",
"3052",
"3277",
"3510",
"3751",
"4000",
"4257",
"4522",
"4795",
"5076",
"5365",
"5662",
"5967",
"6280",
"6601",
"6930",
"7267",
"7612",
"7965",
"8326"
] | 10-gonal (or decagonal) numbers: a(n) = n*(4*n-3). |
A001108 | [
"0",
"1",
"8",
"49",
"288",
"1681",
"9800",
"57121",
"332928",
"1940449",
"11309768",
"65918161",
"384199200",
"2239277041",
"13051463048",
"76069501249",
"443365544448",
"2584123765441",
"15061377048200",
"87784138523761",
"511643454094368",
"2982076586042449",
"17380816062160328",
"101302819786919521"
] | a(n)-th triangular number is a square: a(n+1) = 6*a(n) - a(n-1) + 2, with a(0) = 0, a(1) = 1. |
A001109 | [
"0",
"1",
"6",
"35",
"204",
"1189",
"6930",
"40391",
"235416",
"1372105",
"7997214",
"46611179",
"271669860",
"1583407981",
"9228778026",
"53789260175",
"313506783024",
"1827251437969",
"10650001844790",
"62072759630771",
"361786555939836",
"2108646576008245",
"12290092900109634",
"71631910824649559"
] | a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0, a(1)=1. |
A001110 | [
"0",
"1",
"36",
"1225",
"41616",
"1413721",
"48024900",
"1631432881",
"55420693056",
"1882672131025",
"63955431761796",
"2172602007770041",
"73804512832419600",
"2507180834294496361",
"85170343853180456676",
"2893284510173841030625"
] | Square triangular numbers: numbers that are both triangular and square. |
A001111 | [
"1",
"1",
"1",
"5",
"6",
"1106",
"208310",
"10374196953"
] | Number of inequivalent Hadamard designs of order 4n. |
A001112 | [
"0",
"1",
"1",
"3",
"4",
"11",
"136",
"283",
"419",
"1121",
"1540",
"38081",
"39621",
"117323",
"156944",
"431211",
"5331476",
"11094163",
"16425639",
"43945441",
"60371080",
"1492851361",
"1553222441",
"4599296243",
"6152518684",
"16904333611",
"209004522016",
"434913377643",
"643917899659"
] | A continued fraction. |
A001113 | [
"2",
"7",
"1",
"8",
"2",
"8",
"1",
"8",
"2",
"8",
"4",
"5",
"9",
"0",
"4",
"5",
"2",
"3",
"5",
"3",
"6",
"0",
"2",
"8",
"7",
"4",
"7",
"1",
"3",
"5",
"2",
"6",
"6",
"2",
"4",
"9",
"7",
"7",
"5",
"7",
"2",
"4",
"7",
"0",
"9",
"3",
"6",
"9",
"9",
"9",
"5",
"9",
"5",
"7",
"4",
"9",
"6",
"6",
"9",
"6",
"7",
"6",
"2",
"7",
"7",
"2",
"4",
"0",
"7",
"6",
"6",
"3",
"0",
"3",
"5",
"3",
"5",
"4",
"7",
"5",
"9",
"4",
"5",
"7",
"1",
"3",
"8",
"2",
"1",
"7",
"8",
"5",
"2",
"5",
"1",
"6",
"6",
"4",
"2",
"7",
"4",
"2",
"7",
"4",
"6"
] | Decimal expansion of e. |
A001114 | [
"2",
"7",
"18",
"28",
"182",
"845",
"904",
"5235",
"36028",
"74713",
"526624",
"977572",
"4709369",
"9959574",
"96696762",
"7724076630",
"35354759457",
"138217852516",
"642742746639",
"1932003059921",
"8174135966290",
"43572900334295",
"260595630738132",
"328627943490763",
"2338298807531952",
"5101901157383418"
] | Increasing blocks of digits of e. |
A001115 | [
"1",
"2",
"3",
"4",
"6",
"9",
"14",
"23",
"38",
"64",
"113",
"200",
"358",
"653",
"1202",
"2223",
"4151",
"7781",
"14659",
"27721",
"52603",
"100084",
"190969",
"365134",
"699617",
"1342923",
"2582172",
"4972385",
"9588933",
"18515328",
"35794987",
"69278386",
"134224480",
"260309786",
"505302925",
"981723316",
"1908898002",
"3714597352",
"7233673969",
"14096361346",
"27487875487"
] | Maximal number of pairwise relatively prime polynomials of degree n over GF(2). |
A001116 | [
"0",
"2",
"6",
"12",
"24",
"40",
"72",
"126",
"240",
"272"
] | Maximal kissing number of an n-dimensional lattice. |
A001117 | [
"1",
"0",
"0",
"6",
"36",
"150",
"540",
"1806",
"5796",
"18150",
"55980",
"171006",
"519156",
"1569750",
"4733820",
"14250606",
"42850116",
"128746950",
"386634060",
"1160688606",
"3483638676",
"10454061750",
"31368476700",
"94118013006",
"282379204836",
"847187946150",
"2541664501740",
"7625194831806"
] | a(n) = 3^n - 3*2^n + 3. |
A001118 | [
"1",
"0",
"0",
"0",
"0",
"120",
"1800",
"16800",
"126000",
"834120",
"5103000",
"29607600",
"165528000",
"901020120",
"4809004200",
"25292030400",
"131542866000",
"678330198120",
"3474971465400",
"17710714165200",
"89904730860000",
"454951508208120",
"2296538629446600"
] | Differences of 0; labeled ordered partitions into 5 parts. |
A001119 | [
"1",
"1",
"2",
"2",
"16",
"54"
] | Number of skew-symmetric Hadamard matrices of order 4n. |
A001120 | [
"1",
"1",
"3",
"8",
"33",
"164",
"985",
"6894",
"55153",
"496376",
"4963761",
"54601370",
"655216441",
"8517813732",
"119249392249",
"1788740883734",
"28619854139745",
"486537520375664",
"8757675366761953",
"166395831968477106",
"3327916639369542121",
"69886249426760384540",
"1537497487388728459881"
] | a(0) = a(1) = 1; for n > 1, a(n) = n*a(n-1) + (-1)^n. |
A001121 | [
"1",
"1",
"2",
"2",
"37",
"722"
] | Number of doubly-regular tournaments of order 4n-1. |
A001122 | [
"3",
"5",
"11",
"13",
"19",
"29",
"37",
"53",
"59",
"61",
"67",
"83",
"101",
"107",
"131",
"139",
"149",
"163",
"173",
"179",
"181",
"197",
"211",
"227",
"269",
"293",
"317",
"347",
"349",
"373",
"379",
"389",
"419",
"421",
"443",
"461",
"467",
"491",
"509",
"523",
"541",
"547",
"557",
"563",
"587",
"613",
"619",
"653",
"659",
"661",
"677",
"701",
"709",
"757",
"773",
"787",
"797"
] | Primes with primitive root 2. |
A001123 | [
"7",
"17",
"31",
"43",
"79",
"89",
"113",
"127",
"137",
"199",
"223",
"233",
"257",
"281",
"283",
"331",
"353",
"401",
"449",
"463",
"487",
"521",
"569",
"571",
"593",
"607",
"617",
"631",
"641",
"691",
"739",
"751",
"809",
"811",
"823",
"857",
"881",
"929",
"953",
"977",
"1013",
"1039",
"1049",
"1063",
"1087",
"1097",
"1193",
"1217"
] | Primes with 3 as smallest primitive root. |
A001124 | [
"23",
"47",
"73",
"97",
"103",
"157",
"167",
"193",
"263",
"277",
"307",
"383",
"397",
"433",
"503",
"577",
"647",
"673",
"683",
"727",
"743",
"863",
"887",
"937",
"967",
"983",
"1033",
"1093",
"1103",
"1153",
"1163",
"1223",
"1367",
"1487",
"1543",
"1583",
"1607",
"1777",
"1823",
"1847",
"1933",
"1993",
"2003",
"2017",
"2063",
"2087",
"2113",
"2203",
"2207"
] | Primes with 5 as smallest primitive root. |
A001125 | [
"41",
"109",
"151",
"229",
"251",
"271",
"367",
"733",
"761",
"971",
"991",
"1069",
"1289",
"1303",
"1429",
"1471",
"1759",
"1789",
"1811",
"1879",
"2411",
"2441",
"2551",
"2749",
"2791",
"3061",
"3079",
"3109",
"3229",
"3251",
"3301",
"3319",
"3967",
"4211",
"4549",
"4721",
"4783",
"4909",
"4931",
"4951",
"5101",
"5167",
"5581",
"5791"
] | Primes with 6 as smallest primitive root. |
A001126 | [
"71",
"239",
"241",
"359",
"431",
"499",
"599",
"601",
"919",
"997",
"1051",
"1181",
"1249",
"1439",
"1609",
"1753",
"2039",
"2089",
"2111",
"2179",
"2251",
"2281",
"2341",
"2591",
"2593",
"2671",
"2711",
"2879",
"3119",
"3121",
"3169",
"3181",
"3457",
"3511",
"3541",
"3719",
"3739",
"3769",
"4271",
"4513",
"4799",
"4801",
"4943",
"5197"
] | Primes with 7 as smallest primitive root. |
A001127 | [
"1",
"2",
"4",
"8",
"16",
"77",
"154",
"605",
"1111",
"2222",
"4444",
"8888",
"17776",
"85547",
"160105",
"661166",
"1322332",
"3654563",
"7309126",
"13528163",
"49710694",
"99312488",
"187733887",
"976071668",
"1842242347",
"9274664828",
"17559329557",
"93151725128",
"175304440267",
"937348843838",
"1775697687577"
] | Trajectory of 1 under map x->x + (x-with-digits-reversed). |
A001128 | [
"2",
"4",
"16",
"976",
"662704",
"269896807264",
"124883600543123110859968",
"108643488775144622666209173128243503963147630528"
] | Reverse digits of previous term and multiply by previous term. |
A001129 | [
"0",
"1",
"1",
"2",
"3",
"5",
"8",
"13",
"39",
"124",
"514",
"836",
"1053",
"4139",
"12815",
"61135",
"104937",
"792517",
"1454698",
"9679838",
"17354310",
"9735140",
"1760750",
"986050",
"621360",
"113815",
"581437",
"1252496",
"7676706",
"13019288",
"94367798",
"178067380",
"173537220",
"106496242",
"265429972",
"522619163"
] | Iccanobif numbers: reverse digits of two previous terms and add. |
A001130 | [
"1",
"1",
"3",
"4",
"6",
"11",
"16",
"23",
"36",
"52",
"71",
"103",
"141",
"197",
"272",
"366",
"482",
"657",
"863",
"1140",
"1489",
"1951",
"2511",
"3241",
"4155",
"5317",
"6782",
"8574",
"10786",
"13645",
"17111",
"21313",
"26631",
"33020",
"41005",
"50640",
"62373",
"76510",
"94089",
"114991",
"140376",
"170970",
"207837",
"251552",
"305342",
"368474",
"444360",
"534692",
"642593",
"770278"
] | Number of graphical basis partitions of 2n. |
A001131 | [
"0",
"1",
"2",
"2",
"3",
"8",
"14",
"20",
"35",
"64",
"122",
"260",
"586",
"1296",
"2708",
"5400",
"10468",
"19888",
"37580",
"71960",
"140612",
"279264",
"560544",
"1133760",
"2310316",
"4750368",
"9876264",
"20788880",
"44282696",
"95241664",
"206150208",
"447470464",
"970862029",
"2100029344"
] | Number of red-black rooted trees with n-1 internal nodes. |
A001132 | [
"7",
"17",
"23",
"31",
"41",
"47",
"71",
"73",
"79",
"89",
"97",
"103",
"113",
"127",
"137",
"151",
"167",
"191",
"193",
"199",
"223",
"233",
"239",
"241",
"257",
"263",
"271",
"281",
"311",
"313",
"337",
"353",
"359",
"367",
"383",
"401",
"409",
"431",
"433",
"439",
"449",
"457",
"463",
"479",
"487",
"503",
"521",
"569",
"577",
"593",
"599"
] | Primes == +-1 (mod 8). |
A001133 | [
"43",
"109",
"157",
"229",
"277",
"283",
"307",
"499",
"643",
"691",
"733",
"739",
"811",
"997",
"1021",
"1051",
"1069",
"1093",
"1459",
"1579",
"1597",
"1627",
"1699",
"1723",
"1789",
"1933",
"2179",
"2203",
"2251",
"2341",
"2347",
"2749",
"2917",
"3163",
"3181",
"3229",
"3259",
"3373",
"4027",
"4339",
"4549",
"4597",
"4651",
"4909",
"5101",
"5197",
"5323",
"5413",
"5437",
"5653",
"6037"
] | Primes p such that the multiplicative order of 2 modulo p is (p-1)/3. |
A001134 | [
"113",
"281",
"353",
"577",
"593",
"617",
"1033",
"1049",
"1097",
"1153",
"1193",
"1201",
"1481",
"1601",
"1889",
"2129",
"2273",
"2393",
"2473",
"3049",
"3089",
"3137",
"3217",
"3313",
"3529",
"3673",
"3833",
"4001",
"4217",
"4289",
"4457",
"4801",
"4817",
"4937",
"5233",
"5393",
"5881",
"6121",
"6521",
"6569",
"6761",
"6793",
"6841",
"7129",
"7481",
"7577",
"7793",
"7817",
"7841",
"8209"
] | Primes p such that the multiplicative order of 2 modulo p is (p-1)/4. |
A001135 | [
"251",
"571",
"971",
"1181",
"1811",
"2011",
"2381",
"2411",
"3221",
"3251",
"3301",
"3821",
"4211",
"4861",
"4931",
"5021",
"5381",
"5861",
"6221",
"6571",
"6581",
"8461",
"8501",
"9091",
"9461",
"10061",
"10211",
"10781",
"11251",
"11701",
"11941",
"12541",
"13171",
"13381",
"13421",
"13781",
"14251",
"15541",
"16091",
"16141",
"16451",
"16661",
"16691",
"16811",
"17291"
] | Primes p such that the multiplicative order of 2 modulo p is (p-1)/5. |
A001136 | [
"31",
"223",
"433",
"439",
"457",
"727",
"919",
"1327",
"1399",
"1423",
"1471",
"1831",
"1999",
"2017",
"2287",
"2383",
"2671",
"2767",
"2791",
"2953",
"3271",
"3343",
"3457",
"3463",
"3607",
"3631",
"3823",
"3889",
"4129",
"4423",
"4519",
"4567",
"4663",
"4729",
"4759",
"5167",
"5449",
"5503",
"5953",
"6007",
"6079",
"6151",
"6217",
"6271",
"6673",
"6961",
"6967",
"7321"
] | Primes p such that the multiplicative order of 2 modulo p is (p-1)/6. |
A001137 | [
"1",
"2",
"2",
"4",
"8",
"16",
"33",
"56",
"90",
"164",
"330",
"688",
"1440",
"3008",
"6291",
"13168",
"27604",
"57896",
"120730",
"248312",
"501464",
"995664",
"1954582",
"3821328",
"7495996",
"14848472",
"29815976",
"60741680",
"125363472",
"261452256",
"549461078",
"1160693056",
"2459679936",
"5221717888"
] | Number of black-rooted red-black trees with n internal nodes. |
A001138 | [
"1",
"0",
"1",
"4",
"6",
"4",
"2",
"8",
"32",
"96",
"256",
"608",
"1268",
"2392",
"4177",
"6720",
"9976",
"14064",
"19882",
"30952",
"59080",
"138096",
"355734",
"929040",
"2380268",
"5940408",
"14466720",
"34499984",
"80786736",
"186018208",
"421400951",
"939336288",
"2060601888",
"4450171328",
"9468023540"
] | Red rooted red-black trees with n internal nodes. |
A001139 | [
"1",
"3",
"21",
"6615",
"64595475"
] | Number of stable feedback shift registers with n stages. |
A001140 | [
"4",
"14",
"1114",
"3114",
"132114",
"1113122114",
"311311222114",
"13211321322114",
"1113122113121113222114",
"31131122211311123113322114",
"132113213221133112132123222114",
"11131221131211132221232112111312111213322114",
"31131122211311123113321112131221123113111231121123222114"
] | Describe the previous term! (method A - initial term is 4). |
A001141 | [
"5",
"15",
"1115",
"3115",
"132115",
"1113122115",
"311311222115",
"13211321322115",
"1113122113121113222115",
"31131122211311123113322115",
"132113213221133112132123222115"
] | Describe the previous term! (method A - initial term is 5). |
A001142 | [
"1",
"1",
"2",
"9",
"96",
"2500",
"162000",
"26471025",
"11014635520",
"11759522374656",
"32406091200000000",
"231627686043080250000",
"4311500661703860387840000",
"209706417310526095716965894400",
"26729809777664965932590782608648192"
] | a(n) = Product_{k=1..n} k^(2k - 1 - n). |
A001143 | [
"6",
"16",
"1116",
"3116",
"132116",
"1113122116",
"311311222116",
"13211321322116",
"1113122113121113222116",
"31131122211311123113322116",
"132113213221133112132123222116"
] | Describe the previous term! (method A - initial term is 6). |
A001144 | [
"1",
"2",
"3",
"4",
"9",
"27",
"512",
"134217728"
] | An exponential function on partitions (next term is 2^512). |
A001145 | [
"7",
"17",
"1117",
"3117",
"132117",
"1113122117",
"311311222117",
"13211321322117",
"1113122113121113222117",
"31131122211311123113322117",
"132113213221133112132123222117"
] | Describe the previous term! (method A - initial term is 7). |
A001146 | [
"2",
"4",
"16",
"256",
"65536",
"4294967296",
"18446744073709551616",
"340282366920938463463374607431768211456",
"115792089237316195423570985008687907853269984665640564039457584007913129639936"
] | a(n) = 2^(2^n). |
A001147 | [
"1",
"1",
"3",
"15",
"105",
"945",
"10395",
"135135",
"2027025",
"34459425",
"654729075",
"13749310575",
"316234143225",
"7905853580625",
"213458046676875",
"6190283353629375",
"191898783962510625",
"6332659870762850625",
"221643095476699771875",
"8200794532637891559375",
"319830986772877770815625"
] | Double factorial of odd numbers: a(n) = (2*n-1)!! = 1*3*5*...*(2*n-1). |
A001148 | [
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1",
"3",
"9",
"7",
"1"
] | Final digit of 3^n. |
A001149 | [
"1",
"2",
"3",
"5",
"8",
"13",
"17",
"26",
"34",
"45",
"54",
"67",
"81",
"97",
"115",
"132",
"153",
"171",
"198",
"228",
"256",
"288",
"323",
"357",
"400",
"439",
"488",
"530",
"581",
"627",
"681",
"732",
"790",
"843",
"908",
"963",
"1029",
"1085",
"1152",
"1213",
"1284",
"1346",
"1418",
"1484",
"1561",
"1630",
"1710",
"1785",
"1867",
"1945",
"2034",
"2116"
] | A self-generating sequence: a(1)=1, a(2)=2, a(n+1) chosen so that a(n+1)-a(n-1) is the first number not obtainable as a(j)-a(i) for 1<=i<j<=n. |
A001150 | [
"3",
"13",
"146",
"40422",
"232328410830",
"2110021709419835241732893678",
"88336965390726143627393089434752334013039840509115817923869114"
] | Number of n-input 2-output switching networks with GL(n,2) acting on the input and S(2) and C(2,2) acting on the output. |
A001151 | [
"8",
"18",
"1118",
"3118",
"132118",
"1113122118",
"311311222118",
"13211321322118",
"1113122113121113222118",
"31131122211311123113322118",
"132113213221133112132123222118"
] | Describe the previous term! (method A - initial term is 8). |
A001152 | [
"4",
"36",
"3178",
"298908192",
"165073828103027338592",
"6487168790978377311010208151738379048817328948"
] | Number of n-input 3-output switching networks with GL(n,2) acting on the input and S(3) and C(2,3) acting on the output. |
A001153 | [
"2",
"3",
"5",
"7",
"17",
"31",
"89",
"127",
"521",
"607",
"1279",
"2281",
"3217",
"4423",
"9689",
"19937",
"23209",
"44497",
"110503",
"132049",
"756839",
"859433",
"3021377",
"6972593",
"24036583",
"25964951",
"30402457",
"32582657",
"42643801",
"43112609"
] | Degrees of primitive irreducible trinomials: n such that 2^n - 1 is a Mersenne prime and x^n + x^k + 1 is a primitive irreducible polynomial over GF(2) for some k with 0 < k < n. |
A001154 | [
"9",
"19",
"1119",
"3119",
"132119",
"1113122119",
"311311222119",
"13211321322119",
"1113122113121113222119",
"31131122211311123113322119",
"132113213221133112132123222119"
] | Describe the previous term! (method A - initial term is 9). |
A001155 | [
"0",
"10",
"1110",
"3110",
"132110",
"1113122110",
"311311222110",
"13211321322110",
"1113122113121113222110",
"31131122211311123113322110",
"132113213221133112132123222110",
"11131221131211132221232112111312111213322110",
"31131122211311123113321112131221123113111231121123222110"
] | Describe the previous term! (method A - initial term is 0). |
A001156 | [
"1",
"1",
"1",
"1",
"2",
"2",
"2",
"2",
"3",
"4",
"4",
"4",
"5",
"6",
"6",
"6",
"8",
"9",
"10",
"10",
"12",
"13",
"14",
"14",
"16",
"19",
"20",
"21",
"23",
"26",
"27",
"28",
"31",
"34",
"37",
"38",
"43",
"46",
"49",
"50",
"55",
"60",
"63",
"66",
"71",
"78",
"81",
"84",
"90",
"98",
"104",
"107",
"116",
"124",
"132",
"135",
"144",
"154",
"163",
"169",
"178",
"192",
"201",
"209",
"220",
"235",
"247",
"256"
] | Number of partitions of n into squares. |
A001157 | [
"1",
"5",
"10",
"21",
"26",
"50",
"50",
"85",
"91",
"130",
"122",
"210",
"170",
"250",
"260",
"341",
"290",
"455",
"362",
"546",
"500",
"610",
"530",
"850",
"651",
"850",
"820",
"1050",
"842",
"1300",
"962",
"1365",
"1220",
"1450",
"1300",
"1911",
"1370",
"1810",
"1700",
"2210",
"1682",
"2500",
"1850",
"2562",
"2366",
"2650",
"2210",
"3410",
"2451",
"3255"
] | a(n) = sigma_2(n): sum of squares of divisors of n. |
A001158 | [
"1",
"9",
"28",
"73",
"126",
"252",
"344",
"585",
"757",
"1134",
"1332",
"2044",
"2198",
"3096",
"3528",
"4681",
"4914",
"6813",
"6860",
"9198",
"9632",
"11988",
"12168",
"16380",
"15751",
"19782",
"20440",
"25112",
"24390",
"31752",
"29792",
"37449",
"37296",
"44226",
"43344",
"55261",
"50654",
"61740",
"61544",
"73710",
"68922",
"86688"
] | sigma_3(n): sum of cubes of divisors of n. |
A001159 | [
"1",
"17",
"82",
"273",
"626",
"1394",
"2402",
"4369",
"6643",
"10642",
"14642",
"22386",
"28562",
"40834",
"51332",
"69905",
"83522",
"112931",
"130322",
"170898",
"196964",
"248914",
"279842",
"358258",
"391251",
"485554",
"538084",
"655746",
"707282",
"872644",
"923522",
"1118481",
"1200644"
] | sigma_4(n): sum of 4th powers of divisors of n. |
A001160 | [
"1",
"33",
"244",
"1057",
"3126",
"8052",
"16808",
"33825",
"59293",
"103158",
"161052",
"257908",
"371294",
"554664",
"762744",
"1082401",
"1419858",
"1956669",
"2476100",
"3304182",
"4101152",
"5314716",
"6436344",
"8253300",
"9768751",
"12252702",
"14408200",
"17766056",
"20511150"
] | sigma_5(n), the sum of the 5th powers of the divisors of n. |
A001161 | [
"0",
"4",
"5",
"9",
"11",
"12",
"13",
"14",
"18",
"19",
"20",
"24",
"25",
"29",
"30",
"34",
"35",
"39",
"41",
"42",
"43",
"46",
"47",
"48",
"51",
"52",
"53",
"56",
"57",
"58",
"61",
"62",
"63",
"66",
"67",
"68",
"71",
"72",
"73",
"76",
"77",
"78",
"80",
"84",
"85",
"89",
"90",
"94",
"95",
"99"
] | Numbers containing an even number of letters. |
A001162 | [
"1",
"2",
"3",
"6",
"7",
"8",
"10",
"15",
"16",
"17",
"21",
"22",
"23",
"26",
"27",
"28",
"31",
"32",
"33",
"36",
"37",
"38",
"40",
"44",
"45",
"49",
"50",
"54",
"55",
"59",
"60",
"64",
"65",
"69",
"70",
"74",
"75",
"79",
"81",
"82",
"83",
"86",
"87",
"88",
"91",
"92",
"93",
"96",
"97",
"98"
] | Numbers containing an odd number of letters. |
A001163 | [
"1",
"1",
"1",
"-139",
"-571",
"163879",
"5246819",
"-534703531",
"-4483131259",
"432261921612371",
"6232523202521089",
"-25834629665134204969",
"-1579029138854919086429",
"746590869962651602203151",
"1511513601028097903631961",
"-8849272268392873147705987190261",
"-142801712490607530608130701097701"
] | Stirling's formula: numerators of asymptotic series for Gamma function. |
A001164 | [
"1",
"12",
"288",
"51840",
"2488320",
"209018880",
"75246796800",
"902961561600",
"86684309913600",
"514904800886784000",
"86504006548979712000",
"13494625021640835072000",
"9716130015581401251840000",
"116593560186976815022080000",
"2798245444487443560529920000",
"299692087104605205332754432000000",
"57540880724084199423888850944000000"
] | Stirling's formula: denominators of asymptotic series for Gamma function. |
A001165 | [
"1",
"1",
"3",
"1",
"1",
"1",
"1",
"1",
"1",
"2",
"3",
"1",
"1",
"2",
"1",
"1",
"2",
"1",
"3",
"1",
"2",
"1",
"4",
"1",
"1",
"1",
"3",
"1",
"2",
"1",
"2",
"1",
"2",
"1",
"3",
"1",
"1",
"2",
"1",
"2",
"1",
"1",
"4",
"1",
"2",
"2",
"1",
"2",
"1",
"1",
"2",
"1",
"1",
"2",
"1",
"1",
"2",
"1",
"1",
"2",
"1",
"1",
"4",
"1",
"1",
"2",
"2",
"1",
"2",
"2",
"1",
"1",
"2",
"1",
"1",
"7",
"3",
"1",
"1",
"2",
"1",
"1",
"3",
"2",
"1",
"1",
"2"
] | Position of first even digit after decimal point in sqrt(n). |
A001166 | [
"1",
"4",
"3",
"11",
"15",
"13",
"17",
"24",
"23",
"73",
"3000",
"11000",
"15000",
"101",
"104",
"103",
"111",
"115",
"113",
"117",
"124",
"123",
"173",
"323",
"373",
"1104",
"1103",
"1111",
"1115",
"1113",
"1117",
"1124",
"1123",
"1173",
"1323",
"1373",
"3323",
"3373",
"11373",
"13323",
"13373",
"17373",
"23323",
"23373",
"73373",
"101123",
"101173",
"101323",
"101373",
"103323",
"103373",
"111373",
"113323",
"113373",
"117373"
] | Smallest natural number requiring n letters in English. |
A001167 | [
"1",
"21",
"21000",
"101",
"121",
"1101",
"1121",
"21121",
"101101",
"101121",
"121121",
"1101121",
"1121121",
"21121121",
"101101121",
"101121121",
"121121121",
"1101121121",
"1121121121",
"21121121121",
"101101121121",
"101121121121",
"121121121121",
"1101121121121",
"1121121121121",
"21121121121121"
] | Smallest natural number requiring n words in English (as spoken in England). |
A001168 | [
"1",
"1",
"2",
"6",
"19",
"63",
"216",
"760",
"2725",
"9910",
"36446",
"135268",
"505861",
"1903890",
"7204874",
"27394666",
"104592937",
"400795844",
"1540820542",
"5940738676",
"22964779660",
"88983512783",
"345532572678",
"1344372335524",
"5239988770268",
"20457802016011",
"79992676367108",
"313224032098244",
"1228088671826973"
] | Number of fixed polyominoes with n cells. |
A001169 | [
"1",
"2",
"6",
"19",
"61",
"196",
"629",
"2017",
"6466",
"20727",
"66441",
"212980",
"682721",
"2188509",
"7015418",
"22488411",
"72088165",
"231083620",
"740754589",
"2374540265",
"7611753682",
"24400004911",
"78215909841",
"250726529556",
"803721298537",
"2576384425157",
"8258779154250",
"26474089989299"
] | Number of board-pile polyominoes with n cells. |
A001170 | [
"1",
"2",
"6",
"19",
"63",
"216",
"760",
"2723",
"9880",
"36168",
"133237",
"492993",
"1829670",
"6804267",
"25336611",
"94416842",
"351989967",
"1312471879",
"4894023222",
"18248301701",
"68036380665",
"253638655582",
"945464013411",
"3523978989671",
"13133649924269"
] | Number of board-pair-pile polyominoes with n cells. |
A001171 | [
"1",
"1",
"4",
"20",
"148",
"1348",
"15104",
"198144",
"2998656",
"51290496",
"979732224",
"20661458688",
"476936766720",
"11959743432960",
"323764901314560",
"9410647116349440",
"292316310979706880",
"9663569062008422400",
"338760229843058688000"
] | From least significant term in expansion of E( tr (X'*X)^n ), X rectangular and Gaussian. Also number of types of sequential n-swap moves for traveling salesman problem. |
A001172 | [
"0",
"6",
"10",
"22",
"34",
"48",
"60",
"78",
"84",
"90",
"114",
"144",
"120",
"168",
"180",
"234",
"246",
"288",
"240",
"210",
"324",
"300",
"360",
"474",
"330",
"528",
"576",
"390",
"462",
"480",
"420",
"570",
"510",
"672",
"792",
"756",
"876",
"714",
"798",
"690",
"1038",
"630",
"1008",
"930",
"780",
"960",
"870",
"924",
"900",
"1134",
"1434",
"840",
"990",
"1302"
] | Smallest even number that is an unordered sum of two odd primes in exactly n ways. |
A001173 | [
"1",
"5",
"52",
"1522",
"145984",
"48464496",
"56141454464",
"229148550030864",
"3333310786076963968",
"174695272746749919580928",
"33301710992539090379269318144",
"23278728241293494533015563325552128",
"60084295633556503802059558812644803074048",
"576025077880237078776946730871618386151571214336"
] | Half the number of binary relations on n unlabeled points. |
A001174 | [
"1",
"2",
"7",
"42",
"582",
"21480",
"2142288",
"575016219",
"415939243032",
"816007449011040",
"4374406209970747314",
"64539836938720749739356",
"2637796735571225009053373136",
"300365896158980530053498490893399"
] | Number of oriented graphs (i.e., digraphs with no bidirected edges) on n unlabeled nodes. Also number of complete digraphs on n unlabeled nodes. Number of antisymmetric relations (i.e., oriented graphs with loops) on n unlabeled nodes is A083670. |
A001175 | [
"1",
"3",
"8",
"6",
"20",
"24",
"16",
"12",
"24",
"60",
"10",
"24",
"28",
"48",
"40",
"24",
"36",
"24",
"18",
"60",
"16",
"30",
"48",
"24",
"100",
"84",
"72",
"48",
"14",
"120",
"30",
"48",
"40",
"36",
"80",
"24",
"76",
"18",
"56",
"60",
"40",
"48",
"88",
"30",
"120",
"48",
"32",
"24",
"112",
"300",
"72",
"84",
"108",
"72",
"20",
"48",
"72",
"42",
"58",
"120",
"60",
"30",
"48",
"96",
"140",
"120",
"136"
] | Pisano periods (or Pisano numbers): period of Fibonacci numbers mod n. |
A001176 | [
"1",
"1",
"2",
"1",
"4",
"2",
"2",
"2",
"2",
"4",
"1",
"2",
"4",
"2",
"2",
"2",
"4",
"2",
"1",
"2",
"2",
"1",
"2",
"2",
"4",
"4",
"2",
"2",
"1",
"2",
"1",
"2",
"2",
"4",
"2",
"2",
"4",
"1",
"2",
"2",
"2",
"2",
"2",
"1",
"2",
"2",
"2",
"2",
"2",
"4",
"2",
"2",
"4",
"2",
"2",
"2",
"2",
"1",
"1",
"2",
"4",
"1",
"2",
"2",
"4",
"2",
"2",
"2",
"2",
"2",
"1",
"2",
"4",
"4",
"2",
"1",
"2",
"2",
"1",
"2",
"2",
"2",
"2",
"2",
"4",
"2",
"2",
"2",
"4",
"2",
"2",
"2",
"2",
"2",
"2",
"2",
"4",
"2",
"2",
"2",
"1",
"2",
"2",
"2",
"2"
] | Number of zeros in fundamental period of Fibonacci numbers mod n. |
A001177 | [
"1",
"3",
"4",
"6",
"5",
"12",
"8",
"6",
"12",
"15",
"10",
"12",
"7",
"24",
"20",
"12",
"9",
"12",
"18",
"30",
"8",
"30",
"24",
"12",
"25",
"21",
"36",
"24",
"14",
"60",
"30",
"24",
"20",
"9",
"40",
"12",
"19",
"18",
"28",
"30",
"20",
"24",
"44",
"30",
"60",
"24",
"16",
"12",
"56",
"75",
"36",
"42",
"27",
"36",
"10",
"24",
"36",
"42",
"58",
"60",
"15",
"30",
"24",
"48",
"35",
"60",
"68",
"18",
"24",
"120"
] | Fibonacci entry points: a(n) = least k >= 1 such that n divides Fibonacci number F_k (=A000045(k)). |
A001178 | [
"0",
"4",
"3",
"2",
"3",
"1",
"2",
"2",
"1",
"2",
"3",
"1",
"3",
"2",
"3",
"1",
"2",
"1",
"2",
"2",
"2",
"2",
"2",
"0",
"3",
"3",
"2",
"2",
"3",
"1",
"2",
"2",
"3",
"2",
"2",
"1",
"3",
"2",
"3",
"2",
"3",
"2",
"3",
"2",
"1",
"2",
"3",
"1",
"3",
"2",
"2",
"3",
"3",
"2",
"3",
"2",
"2",
"3",
"4",
"1",
"2",
"2",
"2",
"3",
"3",
"1",
"3",
"2",
"2"
] | Fibonacci frequency of n. |
A001179 | [
"0",
"1",
"1",
"1",
"2",
"1",
"1",
"1",
"1",
"2",
"2",
"1",
"1",
"1",
"2",
"1",
"1",
"1",
"1",
"2",
"1",
"2",
"1",
"1",
"3",
"1",
"1",
"1",
"1",
"2",
"2",
"1",
"2",
"1",
"2",
"1",
"1",
"1",
"1",
"2",
"2",
"1",
"2",
"2",
"2",
"1",
"1",
"1",
"1",
"3",
"1",
"1",
"1",
"1",
"2",
"1",
"1",
"1",
"1",
"2",
"2",
"2",
"1",
"1",
"2",
"2",
"1",
"1",
"1",
"2",
"2",
"1"
] | Leonardo logarithm of n. |
A001180 | [
"1",
"1",
"2",
"3",
"3",
"5",
"9",
"16",
"28",
"50",
"89",
"159",
"285",
"510",
"914",
"1639",
"2938",
"5269",
"9451",
"16952"
] | Erroneous version of A002572. |
A001181 | [
"1",
"1",
"2",
"6",
"22",
"92",
"422",
"2074",
"10754",
"58202",
"326240",
"1882960",
"11140560",
"67329992",
"414499438",
"2593341586",
"16458756586",
"105791986682",
"687782586844",
"4517543071924",
"29949238543316",
"200234184620736",
"1349097425104912",
"9154276618636016",
"62522506583844272"
] | Number of Baxter permutations of length n (also called Baxter numbers). |
A001182 | [
"0",
"1",
"4",
"8",
"15",
"22",
"30",
"41",
"54",
"69",
"83",
"98",
"119",
"139",
"162",
"183",
"208",
"234",
"263",
"294",
"322",
"357",
"390",
"424",
"465",
"504",
"545",
"585",
"628",
"675",
"719",
"770",
"819",
"872",
"928",
"977",
"1036",
"1090",
"1155",
"1216",
"1274",
"1339",
"1404",
"1475",
"1545",
"1610",
"1683",
"1755",
"1832",
"1911",
"1992",
"2072"
] | Number of cells of square lattice of edge 1/n inside quadrant of unit circle centered at 0. |
A001183 | [
"0",
"2",
"2",
"18",
"66",
"374",
"1694",
"9822",
"51698"
] | Number of nontrivial Baxter permutations of length 2n-1. |
A001184 | [
"1",
"2",
"104",
"111712",
"2688307514",
"1445778936756068",
"17337631013706758184626",
"4628650743368437273677525554148",
"27478778338807945303765092195103685118924"
] | Number of simple Hamiltonian paths connecting opposite corners of a 2n+1 X 2n+1 grid. |
A001185 | [
"0",
"1",
"1",
"7",
"21",
"112",
"456",
"2603",
"13203"
] | Number of nontrivial Baxter permutations of length 2n-1. |
A001186 | [
"1",
"2",
"5",
"17",
"80",
"474",
"3841",
"39635",
"495991",
"7170657",
"116171803",
"2070451150",
"40130198979",
"839266928707",
"18826133329753"
] | Number of cubic Hamiltonian graphs with 2n nodes. |
A001187 | [
"1",
"1",
"1",
"4",
"38",
"728",
"26704",
"1866256",
"251548592",
"66296291072",
"34496488594816",
"35641657548953344",
"73354596206766622208",
"301272202649664088951808",
"2471648811030443735290891264",
"40527680937730480234609755344896",
"1328578958335783201008338986845427712"
] | Number of connected labeled graphs with n nodes. |
A001188 | [
"1",
"2",
"8",
"60",
"672",
"9953",
"184557",
"4142631",
"109813842",
"3373122370",
"118280690398",
"4678086540493",
"206625802351035",
"10107719377251109",
"543762148079927802",
"31975474310851749920",
"2044501883873268414092",
"141485408653554069693421"
] | Number of even graphs with n edges. |
A001189 | [
"0",
"1",
"3",
"9",
"25",
"75",
"231",
"763",
"2619",
"9495",
"35695",
"140151",
"568503",
"2390479",
"10349535",
"46206735",
"211799311",
"997313823",
"4809701439",
"23758664095",
"119952692895",
"618884638911",
"3257843882623",
"17492190577599",
"95680443760575",
"532985208200575",
"3020676745975551"
] | Number of degree-n permutations of order exactly 2. |
A001190 | [
"0",
"1",
"1",
"1",
"2",
"3",
"6",
"11",
"23",
"46",
"98",
"207",
"451",
"983",
"2179",
"4850",
"10905",
"24631",
"56011",
"127912",
"293547",
"676157",
"1563372",
"3626149",
"8436379",
"19680277",
"46026618",
"107890609",
"253450711",
"596572387",
"1406818759",
"3323236238",
"7862958391",
"18632325319",
"44214569100"
] | Wedderburn-Etherington numbers: unlabeled binary rooted trees (every node has outdegree 0 or 2) with n endpoints (and 2n-1 nodes in all). |
A001191 | [
"1",
"4",
"9",
"1",
"6",
"2",
"5",
"3",
"6",
"4",
"9",
"6",
"4",
"8",
"1",
"1",
"0",
"0",
"1",
"2",
"1",
"1",
"4",
"4",
"1",
"6",
"9",
"1",
"9",
"6",
"2",
"2",
"5",
"2",
"5",
"6",
"2",
"8",
"9",
"3",
"2",
"4",
"3",
"6",
"1",
"4",
"0",
"0",
"4",
"4",
"1",
"4",
"8",
"4",
"5",
"2",
"9",
"5",
"7",
"6",
"6",
"2",
"5",
"6",
"7",
"6",
"7",
"2",
"9",
"7",
"8",
"4",
"8",
"4",
"1",
"9",
"0",
"0"
] | Digits of positive squares. |
A001192 | [
"1",
"1",
"1",
"2",
"9",
"88",
"1802",
"75598",
"6421599",
"1097780312",
"376516036188",
"258683018091900",
"355735062429124915",
"978786413996934006272",
"5387230452634185460127166",
"59308424712939278997978128490",
"1305926814154452720947815884466579"
] | Number of full sets of size n. |
A001193 | [
"1",
"2",
"9",
"60",
"525",
"5670",
"72765",
"1081080",
"18243225",
"344594250",
"7202019825",
"164991726900",
"4111043861925",
"110681950128750",
"3201870700153125",
"99044533658070000",
"3262279327362680625",
"113987877673731311250",
"4211218814057295665625",
"164015890652757831187500"
] | a(n) = (n+1)*(2*n)!/(2^n*n!) = (n+1)*(2n-1)!!. |
A001194 | [
"3",
"9",
"54",
"450",
"4725",
"59535",
"873180",
"14594580",
"273648375",
"5685805125",
"129636356850",
"3217338674550",
"86331921100425",
"2490343877896875",
"76844896803675000",
"2525635608280785000",
"88081541838792376875",
"3248654513701342370625"
] | a(n) = A059366(n,n-2) = A059366(n,2) for n >= 2, where the triangle A059366 arises in the expansion of a trigonometric integral. |
A001195 | [
"0",
"2",
"4",
"7",
"10",
"13",
"16",
"20",
"24",
"27",
"31",
"36",
"40",
"44",
"48",
"53",
"57",
"62",
"66",
"71",
"76",
"80",
"85",
"90",
"95",
"100",
"105",
"110",
"115",
"120",
"125",
"130",
"136",
"141",
"146",
"152",
"157",
"162",
"168",
"173",
"179",
"184",
"190",
"195",
"201",
"206",
"212",
"218",
"223",
"229"
] | Int(n*log((14/11)*n^(10/9))). |
A001196 | [
"0",
"3",
"12",
"15",
"48",
"51",
"60",
"63",
"192",
"195",
"204",
"207",
"240",
"243",
"252",
"255",
"768",
"771",
"780",
"783",
"816",
"819",
"828",
"831",
"960",
"963",
"972",
"975",
"1008",
"1011",
"1020",
"1023",
"3072",
"3075",
"3084",
"3087",
"3120",
"3123",
"3132",
"3135",
"3264",
"3267",
"3276",
"3279",
"3312",
"3315",
"3324",
"3327",
"3840",
"3843"
] | Double-bitters: only even length runs in binary expansion. |
A001197 | [
"4",
"7",
"10",
"13",
"17",
"22",
"25",
"30",
"35",
"40",
"46",
"53",
"57",
"62",
"68",
"75",
"82",
"89",
"97",
"106",
"109",
"116",
"123"
] | Zarankiewicz's problem k_2(n). |
A001198 | [
"9",
"14",
"21",
"27",
"34",
"43",
"50",
"61",
"70",
"81",
"93",
"106",
"121",
"129"
] | Zarankiewicz's problem k_3(n). |
A001199 | [
"1",
"1",
"2",
"6",
"32",
"353",
"8390",
"436399",
"50468754"
] | Erroneous version of A056642. |
A001200 | [
"1",
"1",
"1",
"2",
"3",
"5",
"10",
"24",
"69",
"384",
"5250",
"232929",
"28872973"
] | Number of linear geometries on n (unlabeled) points. |
Subsets and Splits