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

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

Author

Amiram Eldar, May 17 2024

Keywords

• nonn

Comments

Numbers k such that k | A000172(k).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..345

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.