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A380635 a(1) = 1; a(n+1) = Sum_{d^2|n} a(n/d^2).

%I A380635 #7 Jan 29 2025 07:16:49
%S A380635 1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,5,7,7,8,8,10,10,10,10,12,13,13,14,16,
%T A380635 16,16,16,19,19,19,19,24,24,24,24,28,28,28,28,32,34,34,34,39,40,41,41,
%U A380635 46,46,48,48,53,53,53,53,58,58,58,60,67,67,67,67,74,74,74,74,84,84,84,85,93,93,93,93
%N A380635 a(1) = 1; a(n+1) = Sum_{d^2|n} a(n/d^2).
%F A380635 G.f. A(x) satisfies: A(x) = x * (1 + A(x) + A(x^4) + A(x^9) + ... + A(x^(k^2)) + ...).
%p A380635 a:= proc(n) option remember; uses numtheory; `if`(n=1, 1,
%p A380635       add(`if`(issqr(d), a((n-1)/d), 0), d=divisors(n-1)))
%p A380635     end:
%p A380635 seq(a(n), n=1..80);  # _Alois P. Heinz_, Jan 28 2025
%t A380635 a[1] = 1; a[n_] := a[n] = DivisorSum[n - 1, a[(n - 1)/#] &, IntegerQ[Sqrt[#]] &]; Table[a[n], {n, 1, 80}]
%t A380635 nmax = 80; A[_] = 0; Do[A[x_] = x (1 + Sum[A[x^(k^2)], {k, 1, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
%Y A380635 Cf. A003238, A076752, A167865, A307779, A317240.
%K A380635 nonn
%O A380635 1,5
%A A380635 _Ilya Gutkovskiy_, Jan 28 2025