A016089 -id:A016089 - cp's OEIS frontend

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

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

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