A246692 -id:A246692 - 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);}

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);}

A373056 Numbers k that divide the k-th Ulam number.

Original entry on oeis.org

1, 2, 3, 4, 16, 52, 204, 255, 4259, 4262, 4265, 4855

Offset: 1

Amiram Eldar, May 21 2024

Numbers k such that k | A002858(k).
a(13) >= 10^8, if it exists.
Based on empirical data its seems that the Ulam numbers have a positive asymptotic density and that A002858(k) ~ 13.5... * k (see A307331 and A346216). If this is true, then this sequence is finite, and it is likely that there are no more terms.

			16 is a term since A002858(16) = 48 = 3 * 16 is divisible by 16.

Index entries for Ulam numbers.

Cf. A002858, A307331, A346216.

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

ulams = {1, 2}; Do[AppendTo[ulams, n = Last[ulams]; While[n++; Length[DeleteCases[ Intersection[ulams, n - ulams], n/2, 1, 1]] != 2]; n], {5000}];
Position[ulams/Range[Length[ulams]], ?IntegerQ] // Flatten (* after _Jean-François Alcover at A002858 *)

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

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

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

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