Datasets:
christopher
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oeis

Dataset card Data Studio Files
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Community
2
sequence_id
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7
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573
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348
keywords
sequencelengths
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8
score
int64
1
2.31k
offset_a
int64
-14,827
666,262,453B
offset_b
int64
0
635M
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cross_references
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1
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timestamp
timestamp[us]date
1999-12-11 03:00:00
2025-04-28 00:58:08
filename
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29
29
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stringlengths
32
32
A003001
Smallest number of multiplicative persistence n.
[ "0", "10", "25", "39", "77", "679", "6788", "68889", "2677889", "26888999", "3778888999", "277777788888899" ]
[ "nonn", "nice", "base", "more", "hard" ]
189
0
2
[ "A003001", "A006050", "A007954", "A031286", "A031346", "A031347", "A033908", "A046511", "A121105", "A121111", "A133048", "A133500" ]
[ "M4687" ]
N. J. A. Sloane
2025-03-10T11:50:47
oeisdata/seq/A003/A003001.seq
c93e33a2ceebe57abf5fb8a9c178b960
A003002
Size of the largest subset of the numbers [1...n] which does not contain a 3-term arithmetic progression.
[ "0", "1", "2", "2", "3", "4", "4", "4", "4", "5", "5", "6", "6", "7", "8", "8", "8", "8", "8", "8", "9", "9", "9", "9", "10", "10", "11", "11", "11", "11", "12", "12", "13", "13", "13", "13", "14", "14", "14", "14", "15", "16", "16", "16", "16", "16", "16", "16", "16", "16", "16", "17", "17", "17", "18", "18", "18", "18", "19", "19", "19", "19", "19", "20", "20", "20", "20", "20", "20", "20", "20", "21", "21", "21", "22", "22", "22", "22" ]
[ "nonn", "nice" ]
147
0
3
[ "A003002", "A003003", "A003004", "A003005", "A003278", "A004793", "A005047", "A005487", "A033157", "A065825", "A092482", "A093678", "A093679", "A093680", "A093681", "A093682", "A094870", "A101884", "A101886", "A101888", "A104406", "A140577", "A143824", "A185256", "A208746", "A229037", "A358062" ]
[ "M0275" ]
N. J. A. Sloane
2025-03-22T04:40:31
oeisdata/seq/A003/A003002.seq
579967bb0ca48a13d3e3312cfc458bcf
A003003
Size of the largest subset of the numbers [1...n] which doesn't contain a 4-term arithmetic progression.
[ "1", "2", "3", "3", "4", "5", "5", "6", "7", "8", "8", "8", "9", "9", "10", "10", "11", "11", "12", "12", "13", "13", "14", "14", "15", "15", "16", "17", "17", "18", "18", "18", "19", "20", "20", "20", "21", "21", "21", "22", "22", "22", "23", "23", "24", "24", "24", "25", "25", "26", "26", "26", "27", "28", "28", "28", "28", "29", "29", "30", "30", "30", "30", "31", "31", "32", "32", "33", "33", "34", "34", "34" ]
[ "nonn" ]
64
1
2
[ "A003002", "A003003", "A003004", "A003005", "A003278", "A004793", "A005047", "A005487", "A033157", "A065825", "A092482", "A093678", "A093679", "A093680", "A093681", "A093682", "A094870", "A101884", "A101886", "A101888", "A136746", "A140577", "A185256", "A208746", "A229037" ]
[ "M0439" ]
N. J. A. Sloane
2022-04-13T13:25:16
oeisdata/seq/A003/A003003.seq
2439a05a32b4e05cf0fd746c72df163b
A003004
Size of the largest subset of the numbers [1..n] which does not contain a 5-term arithmetic progression.
[ "1", "2", "3", "4", "4", "5", "6", "7", "8", "8", "9", "10", "11", "12", "12", "13", "14", "15", "16", "16", "16", "16", "16", "17", "18", "18", "19", "20", "21", "21", "22", "22", "23", "24", "24", "25", "26", "27", "28", "28", "29", "30", "31", "32", "32", "32", "32", "32", "33", "33", "34", "35", "35", "36", "36", "37", "38", "39", "40", "40", "41", "42", "43", "44", "44", "45", "46", "47", "48", "48" ]
[ "nonn" ]
44
1
2
[ "A003002", "A003003", "A003004", "A003005", "A065825" ]
[ "M0451" ]
N. J. A. Sloane
2021-12-10T09:41:50
oeisdata/seq/A003/A003004.seq
567b465354b4e123264d3920069f4d58
A003005
Size of the largest subset of the numbers [1..n] which doesn't contain a 6-term arithmetic progression.
[ "1", "2", "3", "4", "5", "5", "6", "7", "8", "9", "9", "10", "11", "12", "13", "13", "14", "15", "16", "17", "17", "18", "19", "20", "21", "22", "22", "22", "23", "23", "23", "24", "25", "25", "26", "27", "28", "28", "29", "30", "31", "31", "31", "32", "33", "34", "34", "35", "36", "37" ]
[ "nonn" ]
45
1
2
[ "A003002", "A003003", "A003004", "A003005", "A065825" ]
[ "M0459" ]
N. J. A. Sloane
2023-10-21T01:07:43
oeisdata/seq/A003/A003005.seq
eb7d4168697d4c623cb381ed3f230358
A003006
Number of n-level ladder expressions with A001622.
[ "1", "1", "2", "3", "7", "15", "35", "81", "195", "473", "1170", "2920", "7378", "18787", "48242", "124658", "324095", "846872", "2223352", "5861011", "15508423", "41173560", "109648734" ]
[ "nonn", "more", "hard" ]
40
1
3
[ "A001622", "A002845", "A003006", "A052321", "A082499", "A198683", "A199812" ]
[ "M0845" ]
N. J. A. Sloane
2025-01-05T10:15:35
oeisdata/seq/A003/A003006.seq
99e04bc8b55effabce106c9bea74e675
A003007
Erroneous version of A082499.
[ "1", "1", "2", "4", "8", "17", "38", "89", "208" ]
[ "dead" ]
28
1
3
null
[ "M1145" ]
null
2025-01-05T02:01:05
oeisdata/seq/A003/A003007.seq
bf829dcc400da1d4e943267be233d39c
A003008
Number of n-level ladder expressions with A030798.
[ "1", "1", "2", "4", "8", "17", "39", "90", "213" ]
[ "nonn", "more" ]
25
1
3
[ "A003008", "A082499" ]
[ "M1147" ]
N. J. A. Sloane
2025-01-05T10:15:01
oeisdata/seq/A003/A003008.seq
43e8efe39e2c82b7aed97080d8eac6c3
A003009
a(n+1) = a(n) + 4^(n-1) a(n)^2.
[ "3", "147", "1383123", "489735485064147", "245597025618959718190041238775763", "247062114274836300381127305147102564467751924522387062291401805739987" ]
[ "nonn", "easy" ]
19
3
1
null
[ "M3174" ]
N. J. A. Sloane
2021-12-26T20:38:39
oeisdata/seq/A003/A003009.seq
03db696d1808ec7f7349fc63ef119fba
A003010
A Lucas-Lehmer sequence: a(0) = 4; for n>0, a(n) = a(n-1)^2 - 2.
[ "4", "14", "194", "37634", "1416317954", "2005956546822746114", "4023861667741036022825635656102100994", "16191462721115671781777559070120513664958590125499158514329308740975788034" ]
[ "nonn", "changed" ]
152
0
1
[ "A001566", "A002812", "A003010", "A003423", "A003487", "A145503" ]
[ "M3494" ]
N. J. A. Sloane
2025-04-14T09:03:06
oeisdata/seq/A003/A003010.seq
491d22bef6f3e48196caf60e3b9a1677
A003011
Number of permutations of up to n kinds of objects, where each kind of object can occur at most two times.
[ "1", "3", "19", "271", "7365", "326011", "21295783", "1924223799", "229714292041", "35007742568755", "6630796801779771", "1527863209528564063", "420814980652048751629", "136526522051229388285611" ]
[ "nonn" ]
36
0
2
[ "A003011", "A082765", "A105748", "A105749" ]
[ "M3071" ]
N. J. A. Sloane
2018-04-10T23:47:00
oeisdata/seq/A003/A003011.seq
9366a0a97615c439bcacbd906f492dd9
A003012
Maximum number of cubes of side length 2 that can be packed into a 3-dimensional torus of side length 2*n+1.
[ "0", "1", "10", "33", "81", "148", "247" ]
[ "nonn", "more" ]
39
0
3
null
[ "M4693" ]
N. J. A. Sloane
2025-03-19T10:05:28
oeisdata/seq/A003/A003012.seq
ec6ae02246d8780db0142ae2a34cac9b
A003013
E.g.f. 1 + x*exp(x) + x^2*exp(2*x).
[ "1", "1", "4", "15", "52", "165", "486", "1351", "3592", "9225", "23050", "56331", "135180", "319501", "745486", "1720335", "3932176", "8912913", "20054034", "44826643", "99614740", "220200981", "484442134", "1061158935", "2315255832", "5033164825" ]
[ "nonn", "easy" ]
32
0
3
[ "A001787", "A001815", "A002999", "A003013", "A005183", "A048482" ]
null
N. J. A. Sloane
2024-08-01T03:05:33
oeisdata/seq/A003/A003013.seq
04e76eb7c74a8bba65de4a94d4cb6694
A003014
Expansion of e.g.f.: 1 + x*exp(x) + x^2*exp(2*x) + x^3*exp(3*x).
[ "1", "1", "4", "21", "124", "705", "3726", "18361", "85240", "376641", "1597690", "6551721", "26116740", "101647585", "387634534", "1452554265", "5360857456", "19523426433", "70272302706", "250318462537", "883418329660", "3091835703201", "10739780397214" ]
[ "nonn", "easy" ]
21
0
3
null
null
N. J. A. Sloane
2025-01-23T16:30:17
oeisdata/seq/A003/A003014.seq
7a7890b675620ae31c9c6ad92142e551
A003015
Numbers that occur 5 or more times in Pascal's triangle.
[ "1", "120", "210", "1540", "3003", "7140", "11628", "24310", "61218182743304701891431482520" ]
[ "nonn" ]
86
1
2
[ "A003015", "A003016", "A059233", "A090162", "A098565", "A182237" ]
[ "M5374" ]
N. J. A. Sloane
2025-02-16T08:32:27
oeisdata/seq/A003/A003015.seq
f2b7c6137a53b0f0cb9a9a0ef44e2d3d
A003016
Number of occurrences of n as an entry in rows <= n of Pascal's triangle (A007318).
[ "0", "3", "1", "2", "2", "2", "3", "2", "2", "2", "4", "2", "2", "2", "2", "4", "2", "2", "2", "2", "3", "4", "2", "2", "2", "2", "2", "2", "4", "2", "2", "2", "2", "2", "2", "4", "4", "2", "2", "2", "2", "2", "2", "2", "2", "4", "2", "2", "2", "2", "2", "2", "2", "2", "2", "4", "4", "2", "2", "2", "2", "2", "2", "2", "2", "2", "4", "2", "2", "2", "3", "2", "2", "2", "2", "2", "2", "2", "4", "2", "2", "2", "2", "2", "4", "2", "2", "2", "2", "2", "2", "4", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2" ]
[ "nonn", "nice", "easy" ]
67
0
2
[ "A003015", "A003016", "A059233", "A138496", "A180058" ]
[ "M0227" ]
N. J. A. Sloane
2025-02-16T08:32:27
oeisdata/seq/A003/A003016.seq
fa6d4d136d2b26feb49009b2f7568e1a
A003017
Expansion of 1/(1 - x*exp(x) + x^2*exp(2*x) - x^3*exp(3*x)).
[ "1", "1", "2", "3", "28", "605", "9366", "116767", "1310408", "15812505", "263924650", "6018875291", "148630870092", "3548481410773", "82543469312318", "2000101425252855", "54347539111582096", "1678367049470539697", "56020955060945897298" ]
[ "nonn" ]
21
0
3
null
null
N. J. A. Sloane
2025-02-19T11:56:46
oeisdata/seq/A003/A003017.seq
959272693f8225320e0135195f1ad289
A003018
Number of distinct values taken by 3^3^...^3 (with n 3's and parentheses inserted in all possible ways).
[ "1", "1", "2", "4", "9", "20", "47", "111", "270", "664", "1659", "4184", "10662", "27367", "70747", "183925", "480656", "1261630", "3324772", "8792592", "23327249", "62067785", "165586565" ]
[ "nonn", "nice", "more" ]
42
1
3
[ "A000081", "A002845", "A003018", "A003019", "A145545", "A145546", "A145547", "A145548", "A145549", "A145550" ]
[ "M1178" ]
N. J. A. Sloane
2018-05-04T23:41:19
oeisdata/seq/A003/A003018.seq
116a1f8533275a34e3ca70679799a53c
A003019
Number of distinct values taken by 4^4^...^4 (with n 4's and parentheses inserted in all possible ways).
[ "1", "1", "2", "4", "9", "20", "48", "114", "282", "703", "1787", "4583", "11900", "31131", "82117", "217954", "581970", "1561704", "4210263", "11396488", "30963024", "84402984", "230779071", "632762424", "1739387089" ]
[ "nonn", "nice", "more" ]
35
1
3
[ "A000081", "A002845", "A003018", "A003019", "A145545", "A145546", "A145547", "A145548", "A145549", "A145550" ]
[ "M1179" ]
N. J. A. Sloane
2022-01-31T01:12:39
oeisdata/seq/A003/A003019.seq
4d53744f86b675248ad370a7734aaca4
A003020
Largest prime factor of the "repunit" number 11...1 (cf. A002275).
[ "11", "37", "101", "271", "37", "4649", "137", "333667", "9091", "513239", "9901", "265371653", "909091", "2906161", "5882353", "5363222357", "333667", "1111111111111111111", "27961", "10838689", "513239", "11111111111111111111111", "99990001", "182521213001", "1058313049" ]
[ "nonn", "nice" ]
70
2
1
[ "A002275", "A003020", "A004023", "A005422", "A006530", "A067063", "A102380" ]
[ "M4790" ]
N. J. A. Sloane
2024-08-01T23:33:50
oeisdata/seq/A003/A003020.seq
e0e164d66c7264e0a53720e048b654b2
A003021
Largest prime factor of 10^n + 1.
[ "2", "11", "101", "13", "137", "9091", "9901", "909091", "5882353", "52579", "27961", "8779", "99990001", "1058313049", "121499449", "9091", "69857", "21993833369", "999999000001", "909090909090909091", "5964848081", "909091", "1056689261", "549797184491917" ]
[ "nonn", "easy" ]
56
0
1
[ "A003021", "A006530", "A062397" ]
[ "M2007" ]
N. J. A. Sloane
2023-10-21T15:52:16
oeisdata/seq/A003/A003021.seq
8cac18ce7de028c5a1ad6d7bc1d63c89
A003022
Length of shortest (or optimal) Golomb ruler with n marks.
[ "1", "3", "6", "11", "17", "25", "34", "44", "55", "72", "85", "106", "127", "151", "177", "199", "216", "246", "283", "333", "356", "372", "425", "480", "492", "553", "585" ]
[ "nonn", "hard", "nice", "more" ]
107
2
2
[ "A003022", "A008404", "A030873", "A036501", "A039953", "A078106", "A103295", "A106683", "A108917", "A143823", "A143824", "A169942", "A227590", "A234943", "A270813", "A325466", "A325545", "A325676", "A325677", "A325678", "A325683" ]
[ "M2540" ]
N. J. A. Sloane
2025-02-16T08:32:27
oeisdata/seq/A003/A003022.seq
349ea58011bbe43656185835246411e6
A003023
"Length" of aliquot sequence for n.
[ "0", "1", "1", "2", "1", "1", "1", "2", "3", "3", "1", "6", "1", "4", "4", "5", "1", "3", "1", "6", "2", "5", "1", "4", "2", "6", "2", "1", "1", "14", "1", "2", "5", "7", "2", "3", "1", "6", "2", "3", "1", "13", "1", "4", "6", "7", "1", "5", "3", "2", "3", "8", "1", "12", "2", "4", "2", "3", "1", "10", "1", "8", "2", "3", "2", "11", "1", "4", "3", "5", "1", "8", "1", "4", "4", "4", "2", "10", "1", "6", "4", "5", "1", "5", "2", "8", "6", "6", "1", "9", "3", "5", "3", "3", "3", "8", "1", "2", "3", "4", "1", "17" ]
[ "nonn", "easy" ]
55
1
4
[ "A001065", "A003023", "A059447", "A098007" ]
[ "M0062" ]
N. J. A. Sloane
2025-02-16T08:32:27
oeisdata/seq/A003/A003023.seq
3264a1531c5305b56474befe293b18e6
A003024
Number of acyclic digraphs (or DAGs) with n labeled nodes.
[ "1", "1", "3", "25", "543", "29281", "3781503", "1138779265", "783702329343", "1213442454842881", "4175098976430598143", "31603459396418917607425", "521939651343829405020504063", "18676600744432035186664816926721", "1439428141044398334941790719839535103", "237725265553410354992180218286376719253505" ]
[ "nonn", "easy", "nice" ]
206
0
3
[ "A003024", "A003025", "A003087", "A003465", "A046860", "A055165", "A058877", "A058891", "A059201", "A081064", "A082402", "A085656", "A086510", "A088957", "A133686", "A135079", "A137435", "A188457", "A188490", "A307049", "A323818", "A334282", "A361718" ]
[ "M3113" ]
N. J. A. Sloane
2025-02-17T12:08:51
oeisdata/seq/A003/A003024.seq
fec19647ca9893e0f1315a7e3adb39cd
A003025
Number of n-node labeled acyclic digraphs with 1 out-point.
[ "1", "2", "15", "316", "16885", "2174586", "654313415", "450179768312", "696979588034313", "2398044825254021110", "18151895792052235541515", "299782788128536523836784628", "10727139906233315197412684689421" ]
[ "nonn" ]
33
1
2
[ "A000169", "A003024", "A003025", "A003026", "A003028", "A003087", "A058876", "A058877", "A058891", "A059201", "A082402", "A088957", "A134531", "A323818", "A334282", "A345258", "A350415", "A350487", "A361718", "A367904", "A368600", "A368601", "A368602" ]
[ "M2083" ]
N. J. A. Sloane
2024-01-04T18:11:22
oeisdata/seq/A003/A003025.seq
926537555b74ea378175a74c92841425
A003026
Number of n-node labeled acyclic digraphs with 2 out-points.
[ "1", "9", "198", "10710", "1384335", "416990763", "286992935964", "444374705175516", "1528973599758889005", "11573608032229769067465", "191141381932394665770442818", "6839625961762363728765713227698" ]
[ "nonn" ]
17
2
2
[ "A003025", "A003026", "A058876" ]
[ "M4666" ]
N. J. A. Sloane
2021-12-27T23:42:18
oeisdata/seq/A003/A003026.seq
a4d127ce57fa61ac02f700e78c1a282a
A003027
Number of weakly connected digraphs with n labeled nodes.
[ "1", "3", "54", "3834", "1027080", "1067308488", "4390480193904", "72022346388181584", "4721717643249254751360", "1237892809110149882059440768", "1298060596773261804821355107253504", "5444502293680983802677246555274553481984", "91343781554246596956424128384394531707099632640" ]
[ "nonn", "easy", "nice" ]
48
1
2
[ "A003027", "A003030", "A003085", "A053763", "A062735" ]
[ "M3161" ]
N. J. A. Sloane
2022-01-11T13:20:13
oeisdata/seq/A003/A003027.seq
87f559bc8905ff81cdca2f24514af0cd
A003028
Number of digraphs on n labeled nodes with a source.
[ "1", "3", "51", "3614", "991930", "1051469032", "4366988803688", "71895397383029040", "4719082081411731363408", "1237678715644664931691596416", "1297992266840866792981316221144960", "5444416466164313011147841248189209354496", "91343356480627224177654291875698256656613808896" ]
[ "nonn", "easy", "nice" ]
35
1
2
[ "A003027", "A003028", "A003029", "A003030", "A049414", "A049524", "A051421", "A057274", "A350792", "A361579" ]
[ "M3157" ]
N. J. A. Sloane
2023-04-14T10:52:39
oeisdata/seq/A003/A003028.seq
7061327361f5ce7a39dac2a0941abf6e
A003029
Number of unilaterally connected digraphs with n labeled nodes.
[ "1", "3", "48", "3400", "955860", "1034141596", "4340689156440", "71756043154026904", "4716284688998245793376", "1237457313794197125403483936", "1297922676419612772784598299454784", "5444329780242560941321703550018388707904", "91342929096228825123968637168641318872709897472" ]
[ "nonn", "nice", "easy" ]
23
1
2
[ "A003027", "A003028", "A003029", "A003030", "A003088", "A049414", "A049524", "A057275" ]
[ "M3156" ]
N. J. A. Sloane
2022-01-19T17:25:25
oeisdata/seq/A003/A003029.seq
7e77f8b783d9f0728bc902ff3059aa06
A003030
Number of strongly connected digraphs with n labeled nodes.
[ "1", "1", "18", "1606", "565080", "734774776", "3523091615568", "63519209389664176", "4400410978376102609280", "1190433705317814685295399296", "1270463864957828799318424676767488", "5381067966826255132459611681511359329536", "90765788839403090457244128951307413371883494400" ]
[ "nonn", "nice" ]
58
1
3
[ "A003030", "A035512", "A054946", "A054947" ]
[ "M5064" ]
N. J. A. Sloane
2024-07-25T11:06:11
oeisdata/seq/A003/A003030.seq
0038e8abbd10fd519c0c99b41154d18f
A003031
Denominators of expansion of Fresnel integral S(z).
[ "6", "336", "42240", "9676800", "3530096640", "1880240947200", "1377317368627200", "1328346084409344000", "1631723190138961920000", "2487305589722682753024000", "4607255276963246207139840000" ]
[ "nonn" ]
22
0
1
null
null
N. J. A. Sloane
2022-09-08T08:44:31
oeisdata/seq/A003/A003031.seq
550d58a8bfcc6865cd7d80490f96ae74
A003032
Smallest integer m such that the product of every 3 consecutive integers > m has a prime factor > prime(n).
[ "2", "8", "48", "98", "350", "440", "2430", "2430", "13310", "13454", "17575", "212380", "212380", "212380", "1205644", "1205644", "1205644", "2018978", "3939648", "3939648", "15473808", "15473808", "407498958", "407498958", "407498958", "138982582998", "138982582998", "138982582998", "138982582998" ]
[ "nonn", "nice" ]
30
2
1
null
[ "M1875" ]
N. J. A. Sloane
2022-01-31T01:12:49
oeisdata/seq/A003/A003032.seq
f70a93c3437e3ff7f78adc6041622539
A003033
Smallest integer m such that the product of every 4 consecutive integers > m has a prime factor > prime(n).
[ "3", "7", "9", "63", "63", "168", "322", "322", "1518", "1518", "1680", "10878", "17575", "17575", "17575", "17575", "17575", "17575", "70224", "70224", "97524", "97524", "97524", "97524", "224846", "224846", "612360", "612360", "15473807", "15473807", "15473807", "15473807", "15473807", "15473807", "15473807", "61011223" ]
[ "nonn" ]
29
3
1
null
[ "M2617" ]
N. J. A. Sloane, Robert G. Wilson v
2015-01-23T02:37:39
oeisdata/seq/A003/A003033.seq
933c45c6a7178d72a0ded63d200ec8ff
A003034
Sylvester's problem: minimal number of ordinary lines through n points in the plane.
[ "3", "3", "4", "3", "3", "4", "6", "5", "6", "6", "6", "7" ]
[ "nonn", "hard", "nice" ]
25
3
1
null
[ "M2275" ]
N. J. A. Sloane
2022-01-01T08:07:27
oeisdata/seq/A003/A003034.seq
62d16d81853815c6268f8002cb4c78b0
A003035
Maximal number of 3-tree rows in n-tree orchard problem.
[ "0", "0", "1", "1", "2", "4", "6", "7", "10", "12", "16", "19", "22", "26" ]
[ "nonn", "nice", "hard", "more" ]
116
1
5
[ "A003035", "A006065", "A008997", "A058212" ]
[ "M0982" ]
N. J. A. Sloane
2025-02-16T08:32:27
oeisdata/seq/A003/A003035.seq
ec56c8961f5ff8fb6447ec0c3f534129
A003036
Number of simplicial arrangements of n lines in the plane (the lines do not pass through a common point, all cells are triangles).
[ "1", "1", "1", "2", "2", "2", "2", "4", "2", "4", "5", "5", "6" ]
[ "nonn", "nice", "more" ]
21
3
4
[ "A003036", "A048872", "A048873" ]
[ "M0223" ]
N. J. A. Sloane
2023-11-15T06:02:37
oeisdata/seq/A003/A003036.seq
8f1aabbe5c4a8a198055033d04479296
A003037
Smallest number of complexity n: smallest number requiring n 1's to build using +, * and ^.
[ "1", "2", "3", "4", "5", "7", "11", "13", "21", "23", "41", "43", "71", "94", "139", "211", "215", "431", "863", "1437", "1868", "2855", "5737", "8935", "15838", "15839", "54357", "95597", "139117", "233195", "470399", "1228247", "2183791", "4388063", "6945587", "13431919", "32329439", "46551023" ]
[ "nonn", "nice" ]
41
1
2
[ "A003037", "A005245", "A005421", "A005520", "A025280", "A117618" ]
[ "M0527" ]
N. J. A. Sloane
2024-08-01T03:10:27
oeisdata/seq/A003/A003037.seq
3877384bbc2574d368e67cec87c101af
A003038
Dimensions of split simple Lie algebras over any field of characteristic zero.
[ "3", "8", "10", "14", "15", "21", "24", "28", "35", "36", "45", "48", "52", "55", "63", "66", "78", "80", "91", "99", "105", "120", "133", "136", "143", "153", "168", "171", "190", "195", "210", "224", "231", "248", "253", "255", "276", "288", "300", "323", "325", "351", "360", "378", "399", "406", "435", "440", "465", "483", "496", "528", "561", "575", "595", "624", "630" ]
[ "nonn", "nice", "easy" ]
23
1
1
[ "A000217", "A000384", "A001066", "A003038", "A005563", "A014105", "A126581" ]
[ "M2712" ]
N. J. A. Sloane
2022-01-31T01:13:04
oeisdata/seq/A003/A003038.seq
09cda8c71f77a04d3f3c26837f35e69e
A003039
Maximal number of prime implicants of a Boolean function of n variables.
[ "1", "2", "6", "13", "32", "92" ]
[ "nonn", "hard", "more", "nice" ]
30
1
2
null
[ "M1596" ]
N. J. A. Sloane
2017-06-09T20:41:06
oeisdata/seq/A003/A003039.seq
d628636af85ccf95da1a7e4d7e5f5594
A003040
Highest degree of an irreducible representation of symmetric group S_n of degree n.
[ "1", "1", "2", "3", "6", "16", "35", "90", "216", "768", "2310", "7700", "21450", "69498", "292864", "1153152", "4873050", "16336320", "64664600", "249420600", "1118939184", "5462865408", "28542158568", "117487079424", "547591590000", "2474843571200", "12760912164000", "57424104738000", "295284192952320" ]
[ "nonn" ]
55
1
3
[ "A003040", "A003869", "A003870", "A003871", "A003872", "A003873", "A003874", "A003875", "A003876", "A003877", "A117500" ]
[ "M0811" ]
N. J. A. Sloane and Richard Stanley
2024-09-23T08:25:45
oeisdata/seq/A003/A003040.seq
1018fa8517fa952591f0965add041096
A003041
Number of vacuously transitive relations on n nodes up to isomorphism.
[ "2", "7", "24", "92", "388" ]
[ "nonn", "more" ]
23
1
1
[ "A003041", "A347700", "A348240" ]
[ "M1764" ]
N. J. A. Sloane
2023-10-20T22:32:16
oeisdata/seq/A003/A003041.seq
9aff7d71a28b3005f514d4c7bbdd7c11
A003042
Number of directed Hamiltonian cycles (or Gray codes) on n-cube.
[ "1", "2", "12", "2688", "1813091520", "71676427445141767741440" ]
[ "nonn", "nice", "hard", "more" ]
60
1
2
[ "A003042", "A003043", "A006069", "A006070", "A066037", "A091299", "A091302", "A159344", "A180026" ]
[ "M2053" ]
N. J. A. Sloane
2025-02-16T08:32:27
oeisdata/seq/A003/A003042.seq
566f036061cb60f1ee238dc770db628f
A003043
Number of Hamiltonian paths (or Gray codes) on n-cube with a marked starting node.
[ "1", "2", "18", "5712", "5859364320" ]
[ "nonn", "hard", "more" ]
40
1
2
[ "A003042", "A003043", "A006069", "A006070", "A066037", "A091299", "A091302", "A179926" ]
[ "M2112" ]
N. J. A. Sloane
2019-01-04T04:58:38
oeisdata/seq/A003/A003043.seq
57e4b4d39b1c5ec4531e8bfb46fdccba
A003044
For n > 4, a(n) is the least integer > a(n-1) with precisely two representations a(n) = a(i) + a(j), 1 <= i < j < n; and a(n) = n for n=1..4.
[ "1", "2", "3", "4", "5", "6", "8", "10", "12", "15", "17", "19", "29", "31", "33", "43", "44", "47", "51", "54", "58", "68", "69", "78", "79", "86", "95", "99", "110", "113", "117", "133", "134", "135", "145", "151", "156", "159", "173", "180", "183", "193", "197", "204", "211", "229", "232", "236", "239", "243", "250", "256", "264", "270", "281", "284" ]
[ "nonn", "nice" ]
33
1
2
[ "A003044", "A060470" ]
[ "M0506" ]
N. J. A. Sloane
2019-04-21T11:03:14
oeisdata/seq/A003/A003044.seq
4710b954d1b20c2165476e361a9e0449
A003045
a(n) (n>6) is least integer > a(n-1) with precisely three representations a(n) = a(i) + a(j), 1 <= i < j < n, a(n) = n for n=1..6.
[ "1", "2", "3", "4", "5", "6", "7", "8", "10", "12", "14", "16", "19", "21", "23", "25", "30", "44", "46", "48", "50", "55", "65", "73", "74", "77", "84", "86", "91", "95", "97", "114", "122", "123", "126", "133", "142", "144", "148", "162", "171", "173", "175", "190", "197", "205", "208", "216", "227", "232", "243", "244", "258", "265", "272", "278", "280", "291", "298", "309", "314", "326" ]
[ "nonn" ]
24
1
2
[ "A003044", "A003045" ]
[ "M0491" ]
N. J. A. Sloane
2023-10-22T00:19:56
oeisdata/seq/A003/A003045.seq
0781df90baa38ffefd926477a97d2c99
A003046
Product of first n Catalan numbers.
[ "1", "1", "2", "10", "140", "5880", "776160", "332972640", "476150875200", "2315045555222400", "38883505145515430400", "2285805733484270091494400", "475475022233529990271933132800", "353230394017289429773019124357120000", "944693494975599542562153266945656012800000" ]
[ "nonn", "easy", "nice" ]
97
0
3
[ "A000108", "A003046", "A003047", "A005249", "A051575", "A055746", "A067689", "A069640", "A074962" ]
[ "M1987" ]
N. J. A. Sloane
2024-11-27T11:09:04
oeisdata/seq/A003/A003046.seq
cfada1d9a055fb5b9862af529d042640
A003047
a(n) = Catalan(n) * Product_{k = 0..n-1} a(k).
[ "1", "1", "2", "10", "280", "235200", "173859840000", "98238542885683200000000", "32169371027674057560745102540800000000000000000", "3518552669874927170883258602839130084576857453953842493259776000000000000000000000000000000000" ]
[ "easy", "nonn" ]
34
1
3
[ "A000108", "A003046", "A003047", "A028580" ]
[ "M1993" ]
N. J. A. Sloane
2023-10-20T22:34:08
oeisdata/seq/A003/A003047.seq
361c624070fa9aa0a2340013314f02f8
A003048
a(n+1) = n*a(n) - (-1)^n.
[ "1", "2", "3", "10", "39", "196", "1175", "8226", "65807", "592264", "5922639", "65149030", "781788359", "10163248668", "142285481351", "2134282220266", "34148515524255", "580524763912336", "10449445750422047", "198539469258018894" ]
[ "nonn" ]
30
1
2
[ "A002467", "A003048" ]
null
N. J. A. Sloane
2024-05-09T13:26:31
oeisdata/seq/A003/A003048.seq
e3c470670603d5c384df212da2490b06
A003049
Number of connected Eulerian graphs with n unlabeled nodes.
[ "1", "0", "1", "1", "4", "8", "37", "184", "1782", "31026", "1148626", "86539128", "12798435868", "3620169692289", "1940367005824561", "1965937435288738165", "3766548132138130650270", "13666503289976224080346733" ]
[ "nonn", "nice", "easy" ]
79
1
5
[ "A002854", "A003049", "A008683" ]
[ "M3344" ]
N. J. A. Sloane
2025-02-16T08:32:27
oeisdata/seq/A003/A003049.seq
0b8a3556b333dd518dc4267a8fd7033e
A003050
Number of primitive sublattices of index n in hexagonal lattice: triples x,y,z from Z/nZ with x+y+z = 0, discarding any triple that can be obtained from another by multiplying by a unit and permuting.
[ "1", "1", "2", "2", "2", "3", "3", "4", "3", "4", "3", "6", "4", "5", "6", "6", "4", "7", "5", "8", "8", "7", "5", "12", "6", "8", "7", "10", "6", "14", "7", "10", "10", "10", "10", "14", "8", "11", "12", "16", "8", "18", "9", "14", "14", "13", "9", "20", "11", "16", "14", "16", "10", "19", "14", "20", "16", "16", "11", "28", "12", "17", "18", "18", "16", "26", "13", "20", "18", "26", "13", "28" ]
[ "nonn", "nice" ]
63
1
3
[ "A001615", "A003050", "A003051", "A006984", "A007997", "A048259", "A054345", "A154272", "A157235" ]
[ "M0229" ]
N. J. A. Sloane
2023-09-19T08:20:25
oeisdata/seq/A003/A003050.seq
e1301033b8b47f3ae6f8b191b803fd0c
A003051
Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are equivalent if they are related by a rotation or reflection preserving the hexagonal lattice.
[ "1", "1", "2", "3", "2", "3", "3", "5", "4", "4", "3", "8", "4", "5", "6", "9", "4", "8", "5", "10", "8", "7", "5", "15", "7", "8", "9", "13", "6", "14", "7", "15", "10", "10", "10", "20", "8", "11", "12", "20", "8", "18", "9", "17", "16", "13", "9", "28", "12", "17", "14", "20", "10", "22", "14", "25", "16", "16", "11", "34", "12", "17", "21", "27", "16", "26", "13", "24", "18", "26", "13", "40", "14" ]
[ "nonn", "nice", "easy" ]
86
1
3
[ "A000203", "A001615", "A003050", "A003051", "A006984", "A054345", "A054346", "A054384", "A069734", "A112689", "A145391", "A145392", "A145393", "A145394", "A159842" ]
[ "M0420" ]
N. J. A. Sloane
2024-09-17T12:44:36
oeisdata/seq/A003/A003051.seq
51508a35178731863790dbcc344ec678
A003052
Self numbers or Colombian numbers (numbers that are not of the form m + sum of digits of m for any m).
[ "1", "3", "5", "7", "9", "20", "31", "42", "53", "64", "75", "86", "97", "108", "110", "121", "132", "143", "154", "165", "176", "187", "198", "209", "211", "222", "233", "244", "255", "266", "277", "288", "299", "310", "312", "323", "334", "345", "356", "367", "378", "389", "400", "411", "413", "424", "435", "446", "457", "468", "479", "490", "501", "512", "514", "525" ]
[ "nonn", "base" ]
169
1
2
[ "A003052", "A006378", "A006886", "A010061", "A055642", "A062028", "A176995", "A232229", "A247104", "A282711", "A283002", "A377472", "A377474" ]
[ "M2404" ]
N. J. A. Sloane
2025-03-11T08:43:06
oeisdata/seq/A003/A003052.seq
c4cd8ff48f9fb5bcaf96c1570a5b8f9d
A003053
Order of orthogonal group O(n, GF(2)).
[ "1", "2", "6", "48", "720", "23040", "1451520", "185794560", "47377612800", "24257337753600", "24815256521932800", "50821645356918374400", "208114637736580743168000", "1704875112338069448032256000", "27930968965434591767112450048000", "915241991059360703024740763172864000" ]
[ "nonn", "easy" ]
91
1
2
[ "A003053", "A003923", "A090770" ]
[ "M1716" ]
N. J. A. Sloane
2024-06-14T22:32:08
oeisdata/seq/A003/A003053.seq
7663349bad084908f7cbc4e783be410e
A003054
Erroneous version of A001011 ("folding a strip of stamps").
[ "1", "1", "2", "5", "14", "39", "120", "358", "1176", "3527", "11622", "36627", "121622", "389560", "1301140", "4215748" ]
[ "dead" ]
14
1
3
null
null
null
2015-07-21T12:48:14
oeisdata/seq/A003/A003054.seq
1dc20ab9e3ee59a4cee1b4d4458d34f0
A003055
Number of connected graphs, up to homeomorphism, that can be drawn in the plane using unit-length edges.
[ "1", "1", "3", "5", "10", "19", "39", "84", "196", "479" ]
[ "nonn", "more", "nice" ]
47
1
3
[ "A003055", "A066951" ]
[ "M2464" ]
N. J. A. Sloane
2025-02-16T08:32:27
oeisdata/seq/A003/A003055.seq
2d3c042c0029f7c741a5cb2da4591114
A003056
n appears n+1 times. Also the array A(n,k) = n+k (n >= 0, k >= 0) read by antidiagonals. Also inverse of triangular numbers.
[ "0", "1", "1", "2", "2", "2", "3", "3", "3", "3", "4", "4", "4", "4", "4", "5", "5", "5", "5", "5", "5", "6", "6", "6", "6", "6", "6", "6", "7", "7", "7", "7", "7", "7", "7", "7", "8", "8", "8", "8", "8", "8", "8", "8", "8", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "11", "11", "11", "11", "11", "11", "11", "11", "11", "11", "11", "11", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12" ]
[ "nonn", "easy", "nice", "tabl" ]
201
0
4
[ "A000196", "A000217", "A000462", "A001227", "A001462", "A001614", "A002024", "A003056", "A004247", "A006463", "A016655", "A048645", "A050600", "A050602", "A073424", "A116939", "A116941", "A116942", "A122797", "A131507", "A238005" ]
null
N. J. A. Sloane
2024-04-24T12:57:04
oeisdata/seq/A003/A003056.seq
4500bfe5b6c4fe9fccf6d1495e3c8dd4
A003057
n appears n - 1 times.
[ "2", "3", "3", "4", "4", "4", "5", "5", "5", "5", "6", "6", "6", "6", "6", "7", "7", "7", "7", "7", "7", "8", "8", "8", "8", "8", "8", "8", "9", "9", "9", "9", "9", "9", "9", "9", "10", "10", "10", "10", "10", "10", "10", "10", "10", "11", "11", "11", "11", "11", "11", "11", "11", "11", "11", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "13", "13", "13", "13", "13", "13", "13", "13", "13", "13", "13", "13", "14", "14", "14", "14" ]
[ "nonn", "easy" ]
75
2
1
[ "A002024", "A002260", "A003057" ]
null
N. J. A. Sloane
2022-07-27T02:20:09
oeisdata/seq/A003/A003057.seq
84f604c6cd34f47c2c854f9ced6e4184
A003058
Duplicate of A000194.
[ "1", "1", "2", "2", "2", "2", "3", "3", "3", "3", "3", "3", "4", "4", "4", "4", "4", "4", "4", "4", "5", "5", "5", "5", "5", "5", "5", "5", "5", "5", "6", "6", "6", "6" ]
[ "dead" ]
7
1
3
null
null
null
2007-05-11T03:00:00
oeisdata/seq/A003/A003058.seq
adb00ff3deb9fbf625e1471d39c116e4
A003059
k appears 2k-1 times. Also, square root of n, rounded up.
[ "1", "2", "2", "2", "3", "3", "3", "3", "3", "4", "4", "4", "4", "4", "4", "4", "5", "5", "5", "5", "5", "5", "5", "5", "5", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "6", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "8", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "9", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10" ]
[ "nonn", "easy" ]
76
1
2
[ "A000196", "A000290", "A002162", "A002522", "A003059", "A010052", "A157466" ]
null
N. J. A. Sloane
2025-02-16T08:32:27
oeisdata/seq/A003/A003059.seq
e9fe2a4c5fbedd85755e368a9226f86c
A003060
Smallest number with reciprocal of period length n in decimal (base 10).
[ "1", "3", "11", "27", "101", "41", "7", "239", "73", "81", "451", "21649", "707", "53", "2629", "31", "17", "2071723", "19", "1111111111111111111", "3541", "43", "23", "11111111111111111111111", "511", "21401", "583", "243", "29", "3191", "211", "2791", "353", "67", "103", "71", "1919", "2028119", "909090909090909091" ]
[ "nonn", "nice" ]
39
0
2
[ "A003060", "A005422", "A007138", "A057951", "A176973", "A212953", "A218356", "A218357", "A218358", "A218359", "A218360", "A218361", "A218362", "A218363", "A218364" ]
[ "M2886" ]
N. J. A. Sloane
2022-07-03T09:16:55
oeisdata/seq/A003/A003060.seq
b5117d17e78943224f3a8659d1a92f6b
A003061
Number of non-Abelian (finite) groups with n conjugacy classes.
[ "0", "0", "1", "2", "7", "7", "11", "18", "24", "36", "34", "49" ]
[ "nonn", "nice", "more", "hard" ]
34
1
4
[ "A000688", "A002319", "A003061", "A006379", "A073043" ]
[ "M1728" ]
N. J. A. Sloane
2025-03-15T15:52:10
oeisdata/seq/A003/A003061.seq
4474fd0d61b258ad9f79d25799d4c9be
A003062
Beginnings of periodic unitary aliquot sequences.
[ "6", "30", "42", "54", "60", "66", "78", "90", "100", "102", "114", "126", "140", "148", "194", "196", "208", "220", "238", "244", "252", "274", "288", "292", "300", "336", "348", "350", "364", "374", "380", "382", "386", "388", "400", "420", "436", "440", "476", "482", "484", "492", "516", "528", "540", "542", "550", "570", "578", "592", "600", "612", "648", "660", "680", "688", "694", "708", "720", "722", "740", "756", "758", "764", "766", "770", "780", "784", "792", "794", "812" ]
[ "nonn" ]
46
1
1
[ "A002827", "A003062", "A034460", "A063991", "A097010", "A097024", "A097030", "A097034", "A097035", "A097036", "A097037", "A318880" ]
[ "M4190" ]
N. J. A. Sloane
2024-01-24T21:28:07
oeisdata/seq/A003/A003062.seq
406460ab515bb193c85c993874e1bcc5
A003063
a(n) = 3^(n-1) - 2^n.
[ "-1", "-1", "1", "11", "49", "179", "601", "1931", "6049", "18659", "57001", "173051", "523249", "1577939", "4750201", "14283371", "42915649", "128878019", "386896201", "1161212891", "3484687249", "10456158899", "31372671001", "94126401611", "282395982049", "847221500579", "2541731610601", "7625329049531", "22876255584049" ]
[ "sign", "easy" ]
46
1
4
[ "A000918", "A003063", "A005173", "A056182", "A064686", "A083313", "A090888", "A214091", "A252750", "A254027", "A363024", "A369490" ]
null
Henrik Johansson (Henrik.Johansson(AT)Nexus.SE)
2024-05-22T12:55:10
oeisdata/seq/A003/A003063.seq
a5d9dce9940fa184ae93cecbfd747b0e
A003064
a(n) = smallest number with shortest addition chain of length n.
[ "1", "2", "3", "5", "7", "11", "19", "29", "47", "71", "127", "191", "379", "607", "1087", "1903", "3583", "6271", "11231", "18287", "34303", "65131", "110591", "196591", "357887", "685951", "1176431", "2211837", "4169527", "7624319", "14143037", "25450463", "46444543", "89209343", "155691199", "298695487", "550040063", "994660991" ]
[ "nonn", "nice", "hard" ]
57
0
2
[ "A003064", "A003065", "A003313", "A075530", "A115617" ]
[ "M0667" ]
N. J. A. Sloane and Don Knuth
2025-04-10T06:46:34
oeisdata/seq/A003/A003064.seq
8bd5e722c121c2406e7f9e0d68d02361
A003065
Number of integers with a shortest addition chain of length n.
[ "1", "1", "2", "3", "5", "9", "15", "26", "44", "78", "136", "246", "432", "772", "1382", "2481", "4490", "8170", "14866", "27128", "49544", "90371", "165432", "303475", "558275", "1028508", "1896704", "3501029", "6465774", "11947258", "22087489", "40886910", "75763102", "140588339", "261070184", "485074788", "901751654", "1677060520" ]
[ "nonn", "nice", "hard" ]
44
0
3
[ "A003064", "A003065", "A003313", "A114623" ]
[ "M0707" ]
N. J. A. Sloane, Don Knuth
2025-04-10T06:46:46
oeisdata/seq/A003/A003065.seq
aff3aaf867a1c088c68f5537c9bc4800
A003066
Problimes (first definition).
[ "2", "4", "6", "9", "12", "15", "19", "23", "27", "31", "35", "40", "45", "50", "55", "60", "65", "70", "75", "80", "86", "92", "98", "104", "110", "116", "122", "128", "134", "140", "146", "152", "158", "164", "170", "176", "182", "188", "194", "200", "206", "213", "220", "227", "234", "241", "248", "255", "262", "269", "276", "283", "290", "297", "304", "311", "318", "325" ]
[ "nonn", "nice" ]
33
1
1
[ "A003066", "A003067", "A003068" ]
[ "M0997" ]
N. J. A. Sloane
2017-07-03T01:15:43
oeisdata/seq/A003/A003066.seq
497f2ea4dde5bb6e2097082759b23b5e
A003067
Problimes (second definition).
[ "2", "4", "7", "10", "13", "17", "21", "25", "29", "34", "39", "44", "49", "54", "59", "64", "69", "74", "79", "84", "90", "96", "102", "108", "114", "120", "126", "132", "138", "144", "150", "156", "162", "168", "174", "180", "186", "192", "198", "204", "210", "216", "222", "228", "235", "242", "249", "256", "263", "270", "277", "284", "291", "298", "305", "312", "319" ]
[ "nonn", "nice" ]
39
1
1
[ "A003066", "A003067", "A003068" ]
[ "M1037" ]
N. J. A. Sloane
2022-04-13T13:25:16
oeisdata/seq/A003/A003067.seq
75858684dadf03d60086d9ae2e858e4e
A003068
Problimes (third definition).
[ "2", "4", "7", "11", "15", "19", "23", "28", "33", "38", "43", "48", "53", "58", "63", "68", "73", "79", "85", "91", "97", "103", "109", "115", "121", "127", "133", "139", "145", "151", "157", "163", "169", "175", "181", "187", "193", "199", "205", "211", "217", "224", "231", "238", "245", "252", "259", "266", "273", "280", "287", "294", "301", "308", "315", "322", "329" ]
[ "nonn", "nice" ]
27
1
1
[ "A003066", "A003067", "A003068" ]
[ "M1038" ]
N. J. A. Sloane
2017-06-09T20:42:29
oeisdata/seq/A003/A003068.seq
cb11d1190eaa97db8d4f6924377a7c22
A003069
Number of n-node digraphs with same converse as complement.
[ "1", "1", "2", "6", "20", "88", "632", "8816", "217616", "10240960", "920250720", "155094492288", "48647483234624", "28422244905194880", "31028639315541654400", "63535600643007773309952", "244919299732387943680047360", "1783414099102940635260044486144" ]
[ "nonn" ]
17
1
3
null
[ "M1525" ]
N. J. A. Sloane
2015-02-06T07:11:25
oeisdata/seq/A003/A003069.seq
c8f94f97840947427f34e5472cbdcaf3
A003070
a(n) = ceiling(log_2 n!).
[ "0", "0", "1", "3", "5", "7", "10", "13", "16", "19", "22", "26", "29", "33", "37", "41", "45", "49", "53", "57", "62", "66", "70", "75", "80", "84", "89", "94", "98", "103", "108", "113", "118", "123", "128", "133", "139", "144", "149", "154", "160", "165", "170", "176", "181", "187", "192", "198", "203", "209", "215", "220", "226", "232", "238", "243", "249", "255", "261", "267" ]
[ "nonn" ]
48
0
4
[ "A003070", "A036604", "A072831" ]
[ "M2407" ]
N. J. A. Sloane
2022-11-04T07:30:51
oeisdata/seq/A003/A003070.seq
8dc0cf987a60fba2f74e7535d4fbb1b9
A003071
Sorting numbers: maximal number of comparisons for sorting n elements by list merging.
[ "0", "1", "3", "5", "9", "11", "14", "17", "25", "27", "30", "33", "38", "41", "45", "49", "65", "67", "70", "73", "78", "81", "85", "89", "98", "101", "105", "109", "115", "119", "124", "129", "161", "163", "166", "169", "174", "177", "181", "185", "194", "197", "201", "205", "211", "215", "220", "225", "242", "245", "249", "253", "259", "263", "268", "273", "283", "287", "292", "297", "304" ]
[ "nonn", "easy", "nice" ]
57
1
3
[ "A001855", "A003071", "A133457" ]
[ "M2443" ]
N. J. A. Sloane
2020-02-05T23:51:23
oeisdata/seq/A003/A003071.seq
72d65f722e5e38c21beb3d3325fe8500
A003072
Numbers that are the sum of 3 positive cubes.
[ "3", "10", "17", "24", "29", "36", "43", "55", "62", "66", "73", "80", "81", "92", "99", "118", "127", "129", "134", "136", "141", "153", "155", "160", "179", "190", "192", "197", "216", "218", "225", "232", "244", "251", "253", "258", "270", "277", "281", "288", "307", "314", "342", "344", "345", "349", "352", "359", "368", "371", "375", "378", "397", "405", "408", "415", "433", "434" ]
[ "nonn", "easy", "nice" ]
83
1
1
[ "A000404", "A000408", "A000414", "A000578", "A003072", "A003325", "A003327", "A003328", "A003329", "A003330", "A003331", "A003332", "A003333", "A003334", "A003335", "A003336", "A003337", "A003338", "A003339", "A003340", "A003341", "A003342", "A003343", "A003344", "A003345", "A003346", "A003347", "A003348", "A003349", "A003350", "A003351", "A003352", "A003353", "A003354", "A003355", "A003356", "A003357", "A003358", "A003359", "A003360", "A003361", "A003362", "A003363", "A003364", "A003365", "A003366", "A003367", "A003368", "A003369", "A003370", "A003371", "A003372", "A003373", "A003374", "A003375", "A003376", "A003377", "A003378", "A003379", "A003380", "A003381", "A003382", "A003383", "A003384", "A003385", "A003387", "A003388", "A003389", "A003390", "A003391", "A003392", "A003393", "A003394", "A003395", "A003396", "A003397", "A003398", "A003399", "A004800", "A004801", "A004802", "A004803", "A004804", "A004805", "A004806", "A004807", "A004808", "A004809", "A004810", "A004811", "A004812", "A004813", "A004814", "A004815", "A004816", "A004817", "A004818", "A004819", "A004820", "A004821", "A004822", "A004823", "A004825", "A010057", "A023042", "A024981", "A047700", "A057904" ]
null
N. J. A. Sloane, David W. Wilson
2025-02-16T08:32:27
oeisdata/seq/A003/A003072.seq
ac4d1cf9bf72c8a2ac361ea871fcb558
A003073
A nonlinear recurrence.
[ "1", "1", "2", "2", "3", "4", "5", "7", "9", "11", "14", "18", "23", "29", "38", "47", "59", "76", "95", "120", "154", "191", "241", "310", "383", "483", "620", "767", "968", "1242", "1535", "1937", "2486", "3071", "3875", "4972", "6143", "7752", "9946", "12287", "15505", "19894", "24575", "31011", "39788", "49151", "62024", "79578", "98303" ]
[ "nonn" ]
19
0
3
null
[ "M0282" ]
N. J. A. Sloane
2022-11-04T07:30:55
oeisdata/seq/A003/A003073.seq
62e20a7395488218790a37aa4c1084fe
A003074
Number of different numbers <= n that are sums of 3 positive cubes.
[ "0", "0", "1", "1", "1", "1", "1", "1", "1", "2", "2", "2", "2", "2", "2", "2", "3", "3", "3", "3", "3", "3", "3", "4", "4", "4", "4", "4", "5", "5", "5", "5", "5", "5", "5", "6", "6", "6", "6", "6", "6", "6", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "7", "8", "8", "8", "8", "8", "8", "8", "9", "9", "9", "9", "10", "10", "10", "10", "10", "10", "10", "11", "11", "11", "11", "11", "11", "11", "12", "13", "13", "13", "13", "13", "13", "13", "13", "13", "13", "13", "14", "14" ]
[ "nonn" ]
9
1
10
[ "A003072", "A003074" ]
null
N. J. A. Sloane, David W. Wilson
2020-09-14T15:23:03
oeisdata/seq/A003/A003074.seq
c1c47be5aba35729363eb9774769151d
A003075
Minimal number of comparisons needed for n-element sorting network.
[ "0", "1", "3", "5", "9", "12", "16", "19", "25", "29", "35", "39" ]
[ "hard", "more", "nonn", "nice" ]
69
1
3
[ "A003075", "A006282", "A036604", "A067782" ]
[ "M2446" ]
N. J. A. Sloane
2022-03-12T22:41:51
oeisdata/seq/A003/A003075.seq
3e1722dcf2ee7e476b407c0625becf9f
A003076
n-th digit after decimal point of square root of n.
[ "0", "0", "1", "2", "0", "6", "9", "3", "2", "0", "1", "5", "7", "9", "4", "6", "0", "4", "6", "2", "1", "8", "6", "3", "8", "0", "0", "4", "2", "4", "8", "5", "9", "9", "0", "4", "0", "0", "3", "2", "1", "2", "4", "3", "9", "8", "9", "0", "1", "0", "4", "3", "4", "3", "5", "7", "0", "8", "4", "3", "7", "7", "5", "7", "0", "6", "2", "6", "0", "5", "7", "0", "0", "3", "6", "4", "3", "0", "2", "5", "9", "0", "3", "8", "8", "8", "4", "0", "1", "8", "5", "4", "7", "3", "9", "6", "0", "7", "0", "2", "0", "6", "6", "5", "8" ]
[ "nonn", "base", "easy" ]
31
0
4
[ "A003076", "A071901" ]
null
N. J. A. Sloane
2024-06-22T22:23:30
oeisdata/seq/A003/A003076.seq
c9934112e5b2df4dd45c7f885e067437
A003077
Decimal expansion of 22/7 - Pi.
[ "0", "0", "1", "2", "6", "4", "4", "8", "9", "2", "6", "7", "3", "4", "9", "6", "1", "8", "6", "8", "0", "2", "1", "3", "7", "5", "9", "5", "7", "7", "6", "3", "9", "9", "7", "2", "9", "4", "5", "6", "8", "7", "7", "4", "3", "4", "8", "2", "0", "3", "7", "0", "3", "6", "1", "6", "7", "9", "1", "2", "5", "5", "0", "5", "4", "9", "3", "2", "6", "4", "5", "0", "8", "5", "6", "6", "4", "8", "1", "4", "4", "2", "2", "9", "1", "0", "8", "0", "3", "1", "8", "0", "0", "7", "4", "0", "0", "7", "4", "8" ]
[ "cons", "nonn" ]
28
0
4
[ "A000796", "A003077", "A068028" ]
null
N. J. A. Sloane
2025-02-16T08:32:27
oeisdata/seq/A003/A003077.seq
5b65aed7de5b4fdcfe5bdd91b9d0bbc2
A003078
Number of letters in n (in Danish).
[ "2", "2", "3", "4", "3", "4", "3", "4", "2", "2", "6", "4", "7", "7", "6", "7", "6", "5", "6", "4", "8", "8", "9", "10", "9", "10", "9", "10", "8", "7", "11", "11", "12", "13", "12", "13", "12", "13", "11", "5", "9", "9", "10", "11", "10", "11", "10", "11", "9", "9", "13", "13", "14", "15", "14", "15", "14", "15", "13", "4", "8", "8", "9", "10", "9", "10", "9", "10", "8", "10", "14" ]
[ "nonn", "word" ]
10
1
1
null
null
Joern Olsson (olsson(AT)math.ku.dk)
2015-01-19T19:22:51
oeisdata/seq/A003/A003078.seq
9194b0a9d5d5da9ff1b1f974fa2128c8
A003079
One of the basic cycles in the x->3x-1 (x odd) or x/2 (x even) problem.
[ "5", "14", "7", "20", "10", "5", "14", "7", "20", "10", "5", "14", "7", "20", "10", "5", "14", "7", "20", "10", "5", "14", "7", "20", "10", "5", "14", "7", "20", "10", "5", "14", "7", "20", "10", "5", "14", "7", "20", "10", "5", "14", "7", "20", "10", "5", "14", "7", "20", "10", "5", "14", "7", "20", "10", "5", "14", "7", "20", "10", "5", "14", "7", "20", "10", "5", "14", "7", "20", "10", "5", "14", "7", "20", "10" ]
[ "nonn", "easy" ]
9
0
1
null
null
N. J. A. Sloane
2023-10-20T22:37:42
oeisdata/seq/A003/A003079.seq
bc95315534a2d131b7f1b7dd18190400
A003080
Number of rooted triangular cacti with 2n+1 nodes (n triangles).
[ "1", "1", "2", "5", "13", "37", "111", "345", "1105", "3624", "12099", "41000", "140647", "487440", "1704115", "6002600", "21282235", "75890812", "272000538", "979310627", "3540297130", "12845634348", "46764904745", "170767429511", "625314778963", "2295635155206", "8447553316546", "31153444946778", "115122389065883" ]
[ "nonn", "eigen", "nice" ]
56
0
3
[ "A003080", "A003081", "A034940", "A034941", "A332648" ]
[ "M1448" ]
N. J. A. Sloane
2021-07-01T03:30:32
oeisdata/seq/A003/A003080.seq
6c18c4409f8c777f72ee96e3b69d543e
A003081
Number of triangular cacti with 2n+1 nodes (n triangles).
[ "1", "1", "1", "2", "4", "8", "19", "48", "126", "355", "1037", "3124", "9676", "30604", "98473", "321572", "1063146", "3552563", "11982142", "40746208", "139573646", "481232759", "1669024720", "5819537836", "20390462732", "71762924354", "253601229046", "899586777908", "3202234779826", "11435967528286", "40964243249727" ]
[ "nonn", "easy", "nice" ]
38
0
4
[ "A003080", "A003081", "A034940", "A034941", "A332649" ]
[ "M1152" ]
N. J. A. Sloane
2020-02-26T09:57:28
oeisdata/seq/A003/A003081.seq
b4ff6970df3ba748fbaffd0310e7790f
A003082
Number of multigraphs with 4 nodes and n edges.
[ "1", "1", "3", "6", "11", "18", "32", "48", "75", "111", "160", "224", "313", "420", "562", "738", "956", "1221", "1550", "1936", "2405", "2958", "3609", "4368", "5260", "6279", "7462", "8814", "10356", "12104", "14093", "16320", "18834", "21645", "24783", "28272", "32158", "36442", "41187", "46410", "52151", "58443", "65345", "72864" ]
[ "easy", "nonn", "nice" ]
49
0
3
[ "A001399", "A003082", "A014395", "A014396", "A014397", "A014398", "A192517", "A290778" ]
[ "M2543" ]
N. J. A. Sloane
2022-11-04T07:31:03
oeisdata/seq/A003/A003082.seq
798c86a3e933a97d2bd510f5797389c0
A003083
Sum a(n) x^n / n = log (1 + Sum g(n) x^n ), where g(n) is # graphs on n nodes (A000088).
[ "1", "3", "7", "27", "106", "681", "5972", "88963", "2349727", "117165818", "11073706216", "1968717966417", "654366802299848", "406048824479878828", "470960717141418629512", "1023512961811602818909395", "4179821138595428450831985657", "32171971054480183600023612728841" ]
[ "nonn", "easy" ]
20
1
2
[ "A000088", "A001349", "A003083" ]
[ "M2691" ]
N. J. A. Sloane
2020-08-11T10:07:02
oeisdata/seq/A003/A003083.seq
68da2408365d3b5844391b77a4d9a9d3
A003084
Related to number of digraphs.
[ "1", "5", "40", "801", "46821", "9185102", "6163297995", "14339791643249", "117235455142196308", "3412474003994007703605", "357748249084029269153547905", "136400554886800212073525651823742", "190697966236731843091458826668123014367", "984418987245772021436902193577676975221669509", "18875177868521443706244256784212908480749407027875180" ]
[ "nonn" ]
25
1
2
null
[ "M3993" ]
N. J. A. Sloane
2021-12-26T20:39:59
oeisdata/seq/A003/A003084.seq
d23b632c0715e5c6bfc69e1793971368
A003085
Number of weakly connected digraphs with n unlabeled nodes.
[ "1", "2", "13", "199", "9364", "1530843", "880471142", "1792473955306", "13026161682466252", "341247400399400765678", "32522568098548115377595264", "11366712907233351006127136886487", "14669074325902449468573755897547924182" ]
[ "nonn", "nice" ]
73
1
2
[ "A000273", "A003027", "A003084", "A003085", "A035512", "A054733", "A350789" ]
[ "M2067" ]
N. J. A. Sloane
2025-02-16T08:32:27
oeisdata/seq/A003/A003085.seq
bb6b1234999d5edcacfb39f42aea4759
A003086
Number of self-complementary digraphs with n nodes.
[ "1", "1", "4", "10", "136", "720", "44224", "703760", "179228736", "9168331776", "9383939974144", "1601371799340544", "6558936236286040064", "3837878966366932639744", "62879572771326489528942592", "128777257564337108286016980992" ]
[ "nonn", "nice", "easy" ]
26
1
3
[ "A000171", "A002785", "A003086" ]
[ "M3404" ]
N. J. A. Sloane
2023-02-20T22:08:38
oeisdata/seq/A003/A003086.seq
aeb2be4c633e2ed465c208d0053cbcb0
A003087
Number of acyclic digraphs with n unlabeled nodes.
[ "1", "1", "2", "6", "31", "302", "5984", "243668", "20286025", "3424938010", "1165948612902", "797561675349580", "1094026876269892596", "3005847365735456265830", "16530851611091131512031070", "181908117707763484218885361402" ]
[ "nonn", "nice" ]
71
0
3
[ "A003024", "A003087", "A082402", "A101228", "A122078", "A350447", "A350448" ]
[ "M1696" ]
N. J. A. Sloane
2025-02-16T08:32:27
oeisdata/seq/A003/A003087.seq
2300a6f04f2a5d3a9131be1e4a0ae0de
A003088
Number of unilateral digraphs with n unlabeled nodes.
[ "1", "1", "2", "11", "171", "8603", "1478644", "870014637" ]
[ "nonn", "nice", "more" ]
23
0
3
[ "A003029", "A003085", "A003088", "A035512", "A049512", "A049531", "A051421", "A057270" ]
[ "M2011" ]
N. J. A. Sloane
2023-02-20T22:08:38
oeisdata/seq/A003/A003088.seq
a555b47520733498c6330abb4f25ffc8
A003089
Number of connected line graphs with n nodes.
[ "1", "1", "2", "5", "12", "30", "79", "227", "710", "2322", "8071", "29503", "112822", "450141", "1867871", "8037472", "35787667", "164551477", "779945969", "3804967442", "19079312775", "98211456209", "518397621443", "2802993986619", "15510781288250", "87765472487659", "507395402140211", "2994893000122118", "18035546081743772", "110741792670074054", "692894304050453139" ]
[ "nonn", "nice" ]
36
1
3
[ "A002905", "A003089" ]
[ "M1417" ]
N. J. A. Sloane
2025-02-16T08:32:27
oeisdata/seq/A003/A003089.seq
b5b9055d29e2b66c7d6c78c6e8f0d385
A003090
Number of species (or "main classes" or "paratopy classes") of Latin squares of order n.
[ "1", "1", "1", "2", "2", "12", "147", "283657", "19270853541", "34817397894749939", "2036029552582883134196099" ]
[ "nonn", "nice", "hard" ]
55
1
4
[ "A000315", "A002860", "A003090", "A040082" ]
[ "M0387" ]
N. J. A. Sloane
2020-02-02T19:55:51
oeisdata/seq/A003/A003090.seq
b4832e9af7a3432f69dc89dd5f048e21
A003091
a(n) = floor( 2^(n*(n-1)/2) / n! ).
[ "1", "1", "1", "2", "8", "45", "416", "6657", "189372", "9695869", "902597327", "154043277297", "48535481831642", "28400190511772276", "31020581422991798557", "63530150754287203445810", "244912468225468597942626507", "1783398168624923337196441201196", "24605638395579573858211783276124626" ]
[ "nonn", "easy" ]
15
1
4
null
[ "M1870" ]
N. J. A. Sloane
2022-11-03T05:46:24
oeisdata/seq/A003/A003091.seq
cf9837d1ad8b55beddd66d9608a67159
A003092
Number of labeled plane 2-trees with n nodes.
[ "0", "1", "2", "36", "1200", "57000", "3477600", "257826240", "22438563840", "2238543216000", "251584613280000", "31431367287936000", "4319334744012288000", "647313578549730892800", "105041172967733882880000", "18345770194541665075200000", "3430869798262479024291840000" ]
[ "nonn" ]
27
1
3
null
[ "M2159" ]
N. J. A. Sloane
2024-02-16T21:09:40
oeisdata/seq/A003/A003092.seq
6ab538d488650d6b083341b0a203b9a8
A003093
Number of planar 2-trees with n nodes.
[ "1", "2", "10", "83", "690", "6412", "61842", "617025", "6302725", "65619820", "693643860", "7425181257", "80326705270", "876832043840", "9645743783040", "106827450870525", "1190150414448795", "13329010510209790", "149976822229984908", "1694617702395672670", "19220402733369657820" ]
[ "nonn" ]
36
3
2
[ "A003092", "A003093", "A120348" ]
[ "M1977" ]
N. J. A. Sloane
2024-02-19T01:35:37
oeisdata/seq/A003/A003093.seq
1f7bbfc835ff985f533520b1754983fc
A003094
Number of unlabeled connected planar simple graphs with n nodes.
[ "1", "1", "1", "2", "6", "20", "99", "646", "5974", "71885", "1052805", "17449299", "313372298", "5942258308" ]
[ "nonn", "nice", "hard", "more", "core" ]
81
0
4
[ "A003094", "A005470", "A049334", "A096332", "A126201" ]
[ "M1652" ]
N. J. A. Sloane
2025-02-16T08:32:27
oeisdata/seq/A003/A003094.seq
35a6e74c45bd70bf6689e8568c6102ab
A003095
a(n) = a(n-1)^2 + 1 for n >= 1, with a(0) = 0.
[ "0", "1", "2", "5", "26", "677", "458330", "210066388901", "44127887745906175987802", "1947270476915296449559703445493848930452791205", "3791862310265926082868235028027893277370233152247388584761734150717768254410341175325352026" ]
[ "nonn", "easy", "nice" ]
189
0
3
[ "A001699", "A003095", "A004019", "A038044", "A056207", "A076949", "A077496", "A091980", "A137560", "A143848", "A143849", "A213437", "A247981", "A248218", "A248219", "A318135", "A318136", "A318137", "A318138", "A318139", "A318140", "A355108" ]
[ "M1544" ]
N. J. A. Sloane and Richard Stanley
2025-01-09T08:28:00
oeisdata/seq/A003/A003095.seq
46483d6b03ec8e170191b3d006b3b79d
A003096
a(n) = a(n-1)^2 - 1, a(0) = 2.
[ "2", "3", "8", "63", "3968", "15745023", "247905749270528", "61457260521381894004129398783", "3776994870793005510047522464634252677140721938309041881088" ]
[ "nonn", "easy", "nice" ]
75
0
1
[ "A003095", "A003096", "A077124", "A139244" ]
[ "M0894" ]
N. J. A. Sloane
2025-01-09T08:28:31
oeisdata/seq/A003/A003096.seq
e65b2ae85af1e5593a49813dda09ac32
A003097
Critical connected topologies with n points.
[ "1", "3", "7", "20", "52", "157", "458", "1436", "4531", "14794", "48944", "165314" ]
[ "nonn", "more" ]
17
2
2
null
[ "M2677" ]
N. J. A. Sloane
2023-10-20T22:40:16
oeisdata/seq/A003/A003097.seq
ebbd05d98ca61b517b210c1680ad3d21
A003098
Palindromic triangular numbers.
[ "0", "1", "3", "6", "55", "66", "171", "595", "666", "3003", "5995", "8778", "15051", "66066", "617716", "828828", "1269621", "1680861", "3544453", "5073705", "5676765", "6295926", "35133153", "61477416", "178727871", "1264114621", "1634004361", "5289009825", "6172882716", "13953435931" ]
[ "nonn", "base", "nice" ]
55
1
3
[ "A000217", "A002113", "A003098", "A008509" ]
[ "M2605" ]
N. J. A. Sloane
2025-03-12T10:37:51
oeisdata/seq/A003/A003098.seq
b5bf3566e9298225600931f61ccc17da
A003099
a(n) = Sum_{k=0..n} binomial(n,k^2).
[ "1", "2", "3", "4", "6", "11", "22", "43", "79", "137", "231", "397", "728", "1444", "3018", "6386", "13278", "26725", "51852", "97243", "177671", "320286", "579371", "1071226", "2053626", "4098627", "8451288", "17742649", "37352435", "77926452", "159899767", "321468048", "632531039", "1219295320", "2308910353", "4314168202" ]
[ "nonn", "easy" ]
55
0
2
[ "A003099", "A010052", "A103198", "A206849" ]
[ "M0576" ]
N. J. A. Sloane, Henry W. Gould
2024-01-22T16:09:23
oeisdata/seq/A003/A003099.seq
921aace6ed130c8613beb134363c25cc
A003100
Decimal Gray code for n.
[ "0", "1", "2", "3", "4", "5", "6", "7", "8", "9", "19", "18", "17", "16", "15", "14", "13", "12", "11", "10", "20", "21", "22", "23", "24", "25", "26", "27", "28", "29", "39", "38", "37", "36", "35", "34", "33", "32", "31", "30", "40", "41", "42", "43", "44", "45", "46", "47", "48", "49", "59", "58", "57", "56", "55", "54", "53", "52", "51", "50", "60", "61", "62", "63", "64", "65", "66", "67", "68", "69", "79", "78", "77" ]
[ "nonn", "base", "easy", "look" ]
43
0
3
[ "A003100", "A174025" ]
[ "M0486" ]
N. J. A. Sloane, Henry W. Gould
2017-06-14T19:22:55
oeisdata/seq/A003/A003100.seq
5dfcc9047633cc7aa67cbc2865b6f28f