cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 19 results. Next

A274518 Numbers k such that k^2 divides A000073(k).

Original entry on oeis.org

1, 103, 112, 2621, 30576, 77168, 694512, 9919728, 24770928, 55638128, 57268848, 80995824, 1300820976
Offset: 1

Views

Author

Seiichi Manyama, Jun 26 2016

Keywords

Examples

			tribonacci(103) = 331800673921785084815380861 = 103^2 * 31275395788649739354829.

Crossrefs

Cf. A000073, A232570.

Programs

  • Ruby
    require 'matrix'
def power(a, n, mod)
  return Matrix.I(a.row_size) if n == 0
  m = power(a, n >> 1, mod)
  m = (m * m).map{|i| i % mod}
  return m if n & 1 == 0
  (m * a).map{|i| i % mod}
end
def f(m, n, mod)
  ary0 = Array.new(m, 0)
  ary0[0] = 1
  v = Vector.elements(ary0)
  ary1 = [Array.new(m, 1)]
  (0..m - 2).each{|i|
    ary2 = Array.new(m, 0)
    ary2[i] = 1
    ary1 << ary2
  }
  a = Matrix[*ary1]
  (power(a, n, mod) * v)[m - 1]
end
p (1..10 ** 6).select{|i| f(3, i, i * i) == 0}

Extensions

a(9)-a(13) from Chai Wah Wu, Jan 29 2018

A299156 Numbers k such that k*(k+1) divides tribonacci(k) (A000073(k)).

Original entry on oeis.org

1, 256, 397, 1197, 8053, 8736, 9901, 32173, 33493, 33757, 38461, 48757, 56101, 57073, 64153, 76561, 79693, 87517, 100608, 102217, 105253, 105601, 105913, 105997, 107713, 108553, 110976, 116293, 123121, 131437, 138517, 143137, 147541, 151237, 156601, 171253
Offset: 1

Views

Author

Seiichi Manyama, Feb 04 2018

Keywords

Comments

A subsequence of A232570.

Examples

			tribonacci(256) = 10285895715599251294835119279496333059462348558276025598603904254464 = 256 * 257 * 156339611436029476149609668037091638184921397104146789862048642.

Links

Crossrefs

Cf. A000073, A217738, A232570, A274518.

Programs

  • Maple
    with(LinearAlgebra[Modular]):
T:= (n, m)-> MatrixPower(m, Mod(m, <<0|1|0>,
    <0|0|1>, <1|1|1>>, float[8]), n)[1, 3]:
a:= proc(n) option remember; local i, k, ok;
      if n=1 then 1 else
        for k from 1+a(n-1) do ok:= true;
          for i in ifactors(k*(k+1))[2] while ok do
            ok:= is(T(k, i[1]^i[2])=0)
          od; if ok then break fi
        od; k
      fi
    end:
seq(a(n), n=1..10);  # Alois P. Heinz, Feb 06 2018
  • Mathematica
    a = b = 0; c = d = 1; k = 2; lst = {1}; While[k < 171255, If[ Mod[c, k (k + 1)] == 0, AppendTo[lst, k]]; a = b; b = c; c = d; d = a + b + c; k++] (* Robert G. Wilson v, Feb 07 2018 *)

A372898 Numbers k that divide the k-th Padovan number.

Original entry on oeis.org

1, 2, 4, 16, 25, 27, 59, 69, 101, 167, 173, 211, 223, 271, 307, 317, 347, 387, 422, 449, 463, 593, 599, 607, 634, 691, 694, 719, 809, 821, 829, 844, 853, 877, 883, 898, 926, 991, 997, 1097, 1117, 1151, 1163, 1181, 1197, 1198, 1231, 1319, 1369, 1388, 1451, 1453, 1481
Offset: 1

Views

Author

Amiram Eldar, May 16 2024

Keywords

Comments

Numbers k such that k | A000931(k).

Examples

			2 is a term since A000931(2) = 0 is divisible by 2.
27 is a term since A000931(27) = 351 = 13 * 27 is divisible by 27.

Links

Crossrefs

Cf. A000931.
Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

Programs

  • Mathematica
    With[{m = 1500}, Position[LinearRecurrence[{0, 1, 1}, {0, 0, 1}, m]/Range[m], _?IntegerQ] // Flatten]
  • PARI
    lista(kmax) = {my(p1 = 0, p2 = 0, p3 = 1, p4); print1("1, 2, "); for(k = 4, kmax, p4 = p1 + p2; if(!(p4 % k), print1(k, ", ")); p1 = p2; p2 = p3; p3 = p4);}

A372899 Numbers k that divide the k-th companion Pell number.

Original entry on oeis.org

1, 2, 6, 18, 54, 66, 162, 198, 486, 594, 726, 1314, 1458, 1782, 2178, 2838, 3222, 3942, 4374, 5346, 5778, 5874, 6534, 7986, 8514, 8646, 9666, 11826, 13122, 14454, 16038, 17334, 17622, 19602, 23958, 25542, 25938, 28998, 31218, 35442, 35478, 39366, 43362, 48114
Offset: 1

Views

Author

Amiram Eldar, May 16 2024

Keywords

Comments

Numbers k such that k | A002203(k).

Examples

			2 is a term since A002203(2) = 6 = 2 * 3 is divisible by 2.
6 is a term since A002203(6) = 198 = 6 * 33 is divisible by 6.

Links

Crossrefs

Cf. A002203.
Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

Programs

  • Mathematica
    Select[Range[50000], Divisible[LucasL[#, 2], #] &]
  • PARI
    lista(kmax) = {my(p1 = 2, p2 = 6, p3); print1("1, 2, "); for(k = 3, kmax, p3 = p1 + 2*p2; if(!(p3 % k), print1(k, ", ")); p1 = p2; p2 = p3);}

A372900 Numbers k that divide the k-th term of Narayana's cows sequence.

Original entry on oeis.org

1, 6, 12, 52, 390, 650, 663, 2077, 11479, 31671, 41158, 43508, 104894, 123682, 127370, 170819, 175075, 191516, 266247, 274378, 327159, 341638, 366903, 383847, 733985, 1236087, 1755063, 1763775, 2277964, 2364654, 3165126, 6726156, 7007823, 7221084, 10903815
Offset: 1

Views

Author

Amiram Eldar, May 16 2024

Keywords

Comments

Numbers k such that k | A000930(k).

Examples

			6 is a term since A000930(6) = 6 is divisible by 6.
12 is a term since A000930(12) = 60 = 5 * 12 is divisible by 12.

Links

Crossrefs

Cf. A000930.
Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

Programs

  • Mathematica
    With[{m = 50000}, Position[LinearRecurrence[{1, 0, 1}, {1, 1, 2}, m]/Range[m], _?IntegerQ] // Flatten]
  • PARI
    lista(kmax) = {my(nc1 = 1, nc2 = 1, nc3 = 2, nc4); print1("1, "); for(k = 4, kmax, nc4 = nc1 + nc3; if(!(nc4 % k), print1(k, ", ")); nc1 = nc2; nc2 = nc3; nc3 = nc4);}

A372901 Numbers k that divide the k-th central Delannoy number.

Original entry on oeis.org

1, 3, 9, 21, 27, 81, 171, 189, 217, 243, 297, 351, 729, 903, 1547, 2187, 3591, 3661, 4131, 5499, 5967, 6019, 6561, 7533, 8001, 11997, 13203, 14217, 15309, 17181, 19683, 20601, 22113, 22599, 23529, 24297, 25659, 26163, 26319, 26487, 28441, 30051, 33021, 37179, 37791
Offset: 1

Views

Author

Amiram Eldar, May 16 2024

Keywords

Comments

Numbers k such that k | A001850(k).

Examples

			3 is a term since A001850(3) = 63 = 3 * 21 is divisible by 3.
9 is a term since A001850(9) = 1462563 = 9 * 162507 is divisible by 9.

Links

Crossrefs

Cf. A001850.
Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

Programs

  • Mathematica
    Select[Range[1000], Divisible[LegendreP[#, 3], #] &]
  • PARI
    lista(kmax) = {my(cd0 = 1, cd1 = 3, cd2); print1("1, "); for(k = 2, kmax, cd2 = (3*(2*k-1)*cd1 - (k-1)*cd0)/k; if(!(cd2 % k), print1(k, ", ")); cd0 = cd1; cd1 = cd2);}

A372902 Numbers k that divide the k-th large Schröder number.

Original entry on oeis.org

1, 2, 6, 33, 42, 154, 198, 258, 270, 342, 850, 1170, 1666, 1806, 2295, 2574, 3262, 3366, 3834, 4070, 4654, 4970, 5439, 6006, 6118, 6162, 6699, 7095, 7254, 7497, 7595, 10241, 11475, 12642, 14014, 15345, 17470, 17670, 18018, 19845, 22446, 23994, 24570, 24651, 25245, 25974, 26334
Offset: 1

Views

Author

Amiram Eldar, May 16 2024

Keywords

Comments

Numbers k such that k | A006318(k).

Examples

			2 is a term since A001850(2) = 6 = 2 * 3 is divisible by 2.
6 is a term since A001850(6) = 1806 = 6 * 301 is divisible by 6.

Links

Crossrefs

Cf. A006318.
Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

Programs

  • Mathematica
    seq[kmax_] := Module[{sc0 = 1, sc1 = 2, sc2, s = {1}}, Do[sc2 = ((6*k-3)*sc1 - (k-2)*sc0)/(k+1); If[Divisible[sc2, k], AppendTo[s, k]]; sc0 = sc1; sc1 = sc2, {k, 2, kmax}]; s]; seq[27000]
  • PARI
    lista(kmax) = {my(sc0 = 1, sc1 = 2, sc2); print1(1, ", "); for(k = 2, kmax, sc2 = ((6*k-3)*sc1 - (k-2)*sc0)/(k+1); if(!(sc2 % k), print1(k, ", ")); sc0 = sc1; sc1 = sc2);}

A372903 Numbers k that divide the k-th little Schroeder number.

Original entry on oeis.org

1, 33, 2295, 5439, 6699, 7095, 7497, 7595, 10241, 11475, 15345, 19845, 24651, 25245, 35845, 37725, 37791, 49203, 50463, 51183, 51471, 60291, 62073, 64337, 65569, 66495, 68313, 78793, 80223, 81809, 86031, 98167, 100659, 103293, 109395, 115245, 119067, 119919, 142137
Offset: 1

Views

Author

Amiram Eldar, May 16 2024

Keywords

Comments

Numbers k such that k | A001003(k).

Examples

			1 is a term since A001003(1) = 2 is divisible by 1.
33 is a term since A001003(33) = 37836272668898230450209 = 33 * 1146553717239340316673 is divisible by 33.

Links

Crossrefs

Cf. A001003.
Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

Programs

  • Mathematica
    seq[kmax_] := Module[{sc0 = 1, sc1 = 1, sc2, s = {1}}, Do[sc2 = ((6*k-3)*sc1 - (k-2)*sc0)/(k+1); If[Divisible[sc2, k], AppendTo[s, k]]; sc0 = sc1; sc1 = sc2, {k, 2, kmax}]; s]; seq[10^5]
  • PARI
    lista(kmax) = {my(sc0 = 1, sc1 = 1, sc2); print1(1, ", "); for(k = 2, kmax, sc2 = ((6*k-3)*sc1 - (k-2)*sc0)/(k+1); if(!(sc2 % k), print1(k, ", ")); sc0 = sc1; sc1 = sc2);}

A372904 Numbers k that divide the k-th central trinomial coefficient.

Original entry on oeis.org

1, 21, 387, 657, 6291, 16113, 25767, 54243, 56457, 96141, 155601, 294273, 300871, 453781, 653421, 660879, 669609, 951881, 993307, 1246077, 1438623, 1535409, 1870533, 2110941, 2510109, 2959173, 2974239, 3158541, 3242673, 3569337, 4139739, 4789273, 5405643, 7034097
Offset: 1

Views

Author

Amiram Eldar, May 16 2024

Keywords

Comments

Numbers k such that k | A002426(k).
Also, numbers k that divide the k-th Riordan number: k | A005043(k).
Apparently a subsequence of A266969.

Examples

			21 is a term since A002426(21) = 1105350729 = 21 * 52635749 is divisible by 21.

Links

Crossrefs

Cf. A002426, A005043.
Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

Programs

  • Mathematica
    Select[Range[1000], Divisible[4^#*JacobiP[#, -# - 1/2, -# - 1/2, -1/2], #] &]
  • PARI
    lista(kmax) = {my(ct0 = 1, ct1 = 1, ct2); print1("1, "); for(k = 2, kmax, ct2 = ((2*k-1)*ct1 + 3*(k-1)*ct0)/k; if(!(ct2 % k), print1(k, ", ")); ct0 = ct1; ct1 = ct2);}

A372940 Numbers k that divide the k-th Franel number.

Original entry on oeis.org

1, 2, 10, 70, 410, 416, 464, 560, 610, 692, 976, 1840, 2512, 2815, 3712, 4187, 5888, 6026, 7192, 10556, 12064, 14560, 18368, 32704, 33580, 36424, 40016, 41944, 45400, 51940, 58115, 60416, 61544, 62930, 64288, 66976, 80320, 87232, 103247, 110026, 114802, 118400
Offset: 1

Views

Author

Amiram Eldar, May 17 2024

Keywords

Comments

Numbers k such that k | A000172(k).

Links

Crossrefs

Cf. A000172.
Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

Programs

  • Mathematica
    seq[kmax_] := Module[{f0 = 1, f1 = 2, f2, s = {1}}, Do[f2 = ((7*k^2 - 7*k + 2)*f1 + 8*(k-1)^2*f0)/k^2; If[Divisible[f2, k], AppendTo[s, k]]; f0 = f1; f1 = f2, {k, 2, kmax}]; s]; seq[5000]
  • PARI
    lista(kmax) = {my(f0 = 1, f1 = 2, f2); print1("1, "); for(k = 2, kmax, f2 = ((7*k^2 - 7*k + 2)*f1 + 8*(k-1)^2*f0)/k^2; if(!(f2 % k), print1(k, ", ")); f0 = f1; f1 = f2);}

Formula

2 is a term since A000172(2) = 10 = 2 * 5 is divisible by 2.
10 is a term since A000172(10) = 38165260 = 10 * 3816526 is divisible by 10.
Showing 1-10 of 19 results. Next

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