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.

Previous Showing 11-20 of 23 results. Next

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.

A372941 Numbers k that divide the k-th Domb number.

Original entry on oeis.org

1, 2, 4, 14, 28, 112, 133, 176, 224, 368, 388, 448, 616, 704, 784, 896, 1216, 1568, 1792, 3563, 4256, 5144, 6272, 8624, 8924, 9856, 11264, 11776, 13927, 16702, 23408, 32936, 38509, 42238, 43456, 43652, 43904, 46424, 67328, 73784, 76912, 78848, 81466, 110614, 118256
Offset: 1

Views

Author

Amiram Eldar, May 17 2024

Keywords

Comments

Numbers k such that k | A002895(k).

Examples

			2 is a term since A002895(2) = 28 = 2 * 14 is divisible by 2.
4 is a term since A002895(4) = 2716 = 4 * 679 is divisible by 4.

Links

Crossrefs

Cf. A002895.
Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

Programs

  • Mathematica
    seq[kmax_] := Module[{d0 = 1, d1 = 4, d2, s = {1}}, Do[d2 = ((20*k^3 - 30*k^2 + 18*k - 4)*d1 - 64*(k-1)^3*d0)/k^3; If[Divisible[d2, k], AppendTo[s, k]]; d0 = d1; d1 = d2, {k, 2, kmax}]; s]; seq[5000]
  • PARI
    lista(kmax) = {my(d0 = 1, d1 = 4, d2); print1("1, "); for(k = 2, kmax, d2 = ((20*k^3 - 30*k^2 + 18*k - 4)*d1 - 64*(k-1)^3*d0)/k^3; if(!(d2 % k), print1(k, ", ")); d0 = d1; d1 = d2);}

A372943 Numbers k that divide the k-th Apéry number (A005258).

Original entry on oeis.org

1, 3, 21, 147, 217, 781, 903, 1323, 3249, 3267, 3591, 5929, 6897, 7623, 8001, 8673, 10017, 11187, 11997, 17181, 21413, 21791, 23529, 38829, 51183, 54033, 58653, 68229, 71391, 75593, 83853, 87813, 97641, 128331, 171647, 217143, 227829, 249159, 302841, 307347, 389403
Offset: 1

Views

Author

Amiram Eldar, May 17 2024

Keywords

Comments

Numbers k such that k | A005258(k).

Examples

			3 is a term since A005258(3) = 147 = 3 * 49 is divisible by 3.

Links

Crossrefs

Cf. A005258.
Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

Programs

  • Mathematica
    seq[kmax_] := Module[{ap0 = 1, ap1 = 3, ap2, s = {1}}, Do[ap2 = ((11*k^2 - 11*k + 3)*ap1 + (k-1)^2*ap0)/k^2; If[Divisible[ap2, k], AppendTo[s, k]]; ap0 = ap1; ap1 = ap2, {k, 2, kmax}]; s]; seq[5000]
  • PARI
    lista(kmax) = {my(ap0 = 1, ap1 = 3, ap2); print1("1, "); for(k = 2, kmax, ap2 = ((11*k^2 - 11*k + 3)*ap1 + (k-1)^2*ap0)/k^2; if(!(ap2 % k), print1(k, ", ")); ap0 = ap1; ap1 = ap2);}

A373054 Numbers k that divide the k-th tetranacci number (A000078).

Original entry on oeis.org

1, 2, 22, 32, 80, 137, 179, 272, 320, 352, 600, 653, 859, 936, 991, 1279, 1280, 1306, 1601, 1609, 1632, 1672, 1982, 2089, 2152, 2437, 2560, 2591, 2693, 2789, 2897, 3120, 3202, 3701, 3823, 3847, 4110, 4212, 4451, 4691, 4751, 4919, 5120, 5182, 5280, 5386, 5431, 5479
Offset: 1

Views

Author

Amiram Eldar, May 20 2024

Keywords

Comments

Numbers k such that k | A000078(k).

Examples

			22 is a term since A000078(22) = 147312 = 22 * 6696 is divisible by 22.

Links

Crossrefs

Cf. A000078.
Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

Programs

  • Mathematica
    With[{m = 10000}, Position[LinearRecurrence[{1, 1, 1, 1}, {0, 0, 1, 1}, m]/Range[m], _?IntegerQ] // Flatten]
  • PARI
    lista(kmax) = {my(t0 = 0, t1 = 0, t2 = 0, t3 = 1, t4 = 0); print1(1, ", ",  2, ", "); for(k = 4, kmax, t4 = t0 + t1 + t2 + t3; if(!(t4%k), print1(k, ", ")); t0 = t1; t1 = t2; t2 = t3; t3 = t4);}

A113471 Lucas(k)/(3k) for k = 2*3^n, where Lucas(k) is k-th Lucas number (A000032).

Original entry on oeis.org

1, 107, 1190741689, 14769352340699478579719327005523, 253650450218391594062880777243777017638488805917392303113120204411172926964476779033181303378188721
Offset: 1

Views

Author

Alexander Adamchuk, May 13 2007

Keywords

Comments

a(n) divides a(n+1). a(n+1)/a(n) = {107, 11128427, 12403489755282666163307, 17174107866559209832245996776509546318861182768126017871860347845227, ...}. a(n+1)/a(n) is prime for n = {1, 2, 4}.

Crossrefs

Cf. A000032, A016089 = numbers n such that n divides n-th Lucas number. Cf. A128935 = Fibonacci(5^n) / 5^n.

Programs

  • Mathematica
    Table[ ( Fibonacci[ 2*3^n - 1 ] + Fibonacci[ 2*3^n + 1 ] ) / ( 2*3^(n+1) ), {n,1,5} ]

Formula

a(n) = ( Fibonacci[ 2*3^n - 1 ] + Fibonacci[ 2*3^n + 1 ] ) / ( 2*3^(n+1) ). a(n) = A000032[ 2*3^n ] / ( 2*3^(n+1) ).

A372942 Numbers k that divide the k-th Apéry number (A005259).

Original entry on oeis.org

1, 5, 55, 629, 3439, 8525, 17629, 74455, 120275, 176305, 244915, 250325, 628975, 817819, 839135, 910675, 912865, 936955, 1118435, 1147925, 2344127, 4434125, 7795715, 7888477, 9276275, 10205525
Offset: 1

Views

Author

Amiram Eldar, May 17 2024

Keywords

Comments

Numbers k such that k | A005259(k).

Crossrefs

Cf. A005259.
Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

Programs

  • Mathematica
    seq[kmax_] := Module[{ap0 = 1, ap1 = 5, ap2, s = {1}}, Do[ap2 = ((34*k^3 - 51*k^2 + 27*k - 5)*ap1 - (k-1)^3*ap0)/k^3; If[Divisible[ap2, k], AppendTo[s, k]]; ap0 = ap1; ap1 = ap2, {k, 2, kmax}]; s]; seq[5000]
  • PARI
    lista(kmax) = {my(ap0 = 1, ap1 = 5, ap2); print1("1, "); for(k = 2, kmax, ap2 = ((34*k^3 - 51*k^2 + 27*k - 5)*ap1 - (k-1)^3*ap0)/k^3; if(!(ap2 % k), print1(k, ", ")); ap0 = ap1; ap1 = ap2);}

Formula

5 is a term since A005259(5) = 819005 = 5 * 163801 is divisible by 5.

A372944 Numbers k that divide the k-th tangent (or "zag") number.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 68, 128, 256, 512, 592, 1024, 1156, 2048, 2056, 4096, 4112, 8192, 8224, 8576, 10928, 16384, 16448, 19652, 20512, 28936, 32768, 37888, 41024, 43882, 64804, 65536, 82048
Offset: 1

Views

Author

Amiram Eldar, May 17 2024

Keywords

Comments

Numbers k such that k | A000182(k).
All the powers of 2 are terms.

Examples

			2 is a term since A000182(2) = 2 is divisible by 2.
4 is a term since A000182(4) = 272 = 4 * 68 is divisible by 4.

Crossrefs

Cf. A000182.
Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

Programs

  • Mathematica
    Select[Range[1000], Divisible[((-4)^# - (-16)^#) * BernoulliB[2*#]/(2*#), #] &]
  • PARI
    is(n) = (((-4)^n - (-16)^n) * bernfrac(2*n) / (2*n)) % n == 0;

A372945 Numbers k that divide the k-th Wedderburn-Etherington number.

Original entry on oeis.org

1, 6, 36, 49, 61, 223, 4258, 9747
Offset: 1

Views

Author

Amiram Eldar, May 17 2024

Keywords

Comments

Numbers k such that k | A001190(k).
a(9) > 90000, if it exists.

Examples

			6 is a term since A001190(6) = 6 is divisible by 6.
36 is a term since A001190(36) = 249959727972 = 36 * 6943325777 is divisible by 36.

Crossrefs

Cf. A001190.
Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

Programs

  • Mathematica
    v[0] = 0; v[1] = 1; v[n_] := v[n] = Sum[v[k] * v[n-k], {k, 1, Floor[(n-1)/2]}] + If[EvenQ[n], v[n/2]*(v[n/2]+1)/2, 0]; Select[Range[10^4], Divisible[v[#], #] &]
  • PARI
    lista(kmax) = {my(v = vector(kmax, i, 1)); print1(1, ", "); for(k = 4, kmax, v[k] = sum(i = 1, (k-1)\2, v[i] * v[k-i]) + if(!(k % 2), v[k/2] * (v[k/2] + 1)/2); if(!(v[k] % k), print1(k, ", ")));}
Previous Showing 11-20 of 23 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.