A051177 -id:A051177 - cp's OEIS frontend

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

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

Amiram Eldar, May 17 2024

nonn

Numbers k such that k | A002895(k).

			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.

Amiram Eldar, Table of n, a(n) for n = 1..132

Cf. A002895.

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

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]

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

Amiram Eldar, May 17 2024

nonn

Numbers k such that k | A005258(k).

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

Amiram Eldar, Table of n, a(n) for n = 1..143

Cf. A005258.

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

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]

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

Amiram Eldar, May 20 2024

nonn

Numbers k such that k | A000078(k).

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000078.

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

With[{m = 10000}, Position[LinearRecurrence[{1, 1, 1, 1}, {0, 0, 1, 1}, m]/Range[m], _?IntegerQ] // Flatten]

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);}

A302752 Numbers k such that k divides the sum of largest parts of all partitions of k.

Original entry on oeis.org

1, 3, 4, 5, 11, 197, 487, 928, 10995, 19318

Offset: 1

Ilya Gutkovskiy, Apr 12 2018

Numbers k such that k divides the total number of parts in all partitions of k.

Next term, if it exists, is greater than 100000. - Vaclav Kotesovec, May 04 2018

			5 is in the sequence because we have [5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1] and 5 + 4 + 3 + 3 + 2 + 2 + 1 = 20 is divisible by 5.

Cf. A006128, A051177.

Select[Range[200],Mod[Total[IntegerPartitions[#][[;;,1]]],#]==0&] (* The program generates the first six terms of the sequence. *) (* Harvey P. Dale, Oct 03 2023 *)

A363252 a(n) = gcd(A000041(n), A000009(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 5, 2, 2, 2, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 2, 3, 2, 2, 1, 4, 2, 3, 7, 2, 3, 1, 1, 1, 1, 21, 21, 2, 1, 1, 2, 6, 14, 3, 1, 2, 1, 1, 1, 1, 2, 1, 3, 4, 4, 17, 1, 2, 1, 2, 2, 4, 1, 3, 5, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 4, 1, 1, 1, 2, 11, 2

Offset: 0

Vaclav Kotesovec, May 23 2023, inspired by Zhi-Wei Sun

nonn

Cf. A000009, A000041, A051177, A056848, A093952, A304991.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> igcd(b(n), combinat[numbpart](n)):
seq(a(n), n=0..120);  # Alois P. Heinz, May 23 2023

Table[GCD[PartitionsP[n], PartitionsQ[n]], {n, 0, 100}]

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

Amiram Eldar, May 17 2024

Numbers k such that k | A005259(k).

Cf. A005259.

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

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]

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);}

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

Amiram Eldar, May 17 2024

Numbers k such that k | A000182(k).

All the powers of 2 are terms.

			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.

Cf. A000182.

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

Select[Range[1000], Divisible[((-4)^# - (-16)^#) * BernoulliB[2*#]/(2*#), #] &]

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

Amiram Eldar, May 17 2024

Numbers k such that k | A001190(k).

a(9) > 90000, if it exists.

			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.

Cf. A001190.

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

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[#], #] &]

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, ", ")));}

A372946 Numbers k that divide the k-th NSW number.

Original entry on oeis.org

1, 7, 217, 3937, 6727, 6847, 51943, 170671, 330337, 385687, 2484247, 2566537, 2904007, 3020857, 3696967, 6465577, 9405337, 12021439, 19384207

Offset: 1

Amiram Eldar, May 17 2024

Numbers k such that k | A002315(k).

			7 is a term since A002315(7) = 275807 = 7 * 39401 is divisible by 7.

Cf. A002315, A330276.

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

seq[kmax_] := Module[{nsw0 = 1, nsw1 = 7, nsw2, s = {1}}, Do[nsw2 = 6*nsw1 - nsw0; If[Divisible[nsw2, k], AppendTo[s, k]]; nsw0 = nsw1; nsw1 = nsw2, {k, 2, kmax}]; s]; seq[52000]

lista(kmax) = {my(nsw0 = 1, nsw1 = 7, nsw2); print1("1, "); for(k = 2, kmax, nsw2 = 6*nsw1 - nsw0; if(!(nsw2 % k), print1(k, ", ")); nsw0 = nsw1; nsw1 = nsw2);}

A373055 Numbers k that divide the k-th term of the tribonacci sequence A000213.

Original entry on oeis.org

1, 3, 217, 13343, 549333, 1387663, 9356863, 22119541

Offset: 1

Amiram Eldar, May 21 2024

Numbers k such that k | A000213(k).

			3 is a term since A000213(3) = 3 is divisible by 3.

Cf. A000213.

Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci A000073), A246692 (Pell), A266969 (Motzkin).

With[{m = 15000}, Position[LinearRecurrence[{1, 1, 1}, {1, 1, 3}, m]/Range[m], _?IntegerQ] // Flatten]

lista(kmax) = {my(t0 = 1, t1 = 1, t2 = 1, t3); print1("1, "); for(k = 3, kmax, t3 = t0 + t1 + t2; if(!(t3 % k), print1(k, ", ")); t0 = t1; t1 = t2; t2 = t3);}

cp's OEIS Frontend

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

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

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

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A302752 Numbers k such that k divides the sum of largest parts of all partitions of k.

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A363252 a(n) = gcd(A000041(n), A000009(n)).

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

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Formula

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

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

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