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%I #11 May 18 2024 02:01:09
%S 1,2,10,70,410,416,464,560,610,692,976,1840,2512,2815,3712,4187,5888,
%T 6026,7192,10556,12064,14560,18368,32704,33580,36424,40016,41944,
%U 45400,51940,58115,60416,61544,62930,64288,66976,80320,87232,103247,110026,114802,118400
%N Numbers k that divide the k-th Franel number.
%C Numbers k such that k | A000172(k).
%H Amiram Eldar, <a href="/A372940/b372940.txt">Table of n, a(n) for n = 1..345</a>
%F 2 is a term since A000172(2) = 10 = 2 * 5 is divisible by 2.
%F 10 is a term since A000172(10) = 38165260 = 10 * 3816526 is divisible by 10.
%t 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]
%o (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);}
%Y Cf. A000172.
%Y Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).
%K nonn
%O 1,2
%A _Amiram Eldar_, May 17 2024
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