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%I #10 May 18 2024 09:52:40
%S 1,6,36,49,61,223,4258,9747
%N Numbers k that divide the k-th Wedderburn-Etherington number.
%C Numbers k such that k | A001190(k).
%C a(9) > 90000, if it exists.
%e 6 is a term since A001190(6) = 6 is divisible by 6.
%e 36 is a term since A001190(36) = 249959727972 = 36 * 6943325777 is divisible by 36.
%t 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[#], #] &]
%o (PARI) 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, ", ")));}
%Y Cf. A001190.
%Y Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).
%K nonn,more
%O 1,2
%A _Amiram Eldar_, May 17 2024