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a(n) = 1 + Sum_{d|n, d < n} a(d - 1).
4

%I #8 Apr 13 2021 07:13:03

%S 1,1,2,2,3,2,5,2,5,4,6,2,9,2,8,7,7,2,12,2,12,9,9,2,13,5,12,9,12,2,22,

%T 2,14,10,10,10,18,2,15,13,16,2,26,2,20,20,12,2,22,7,23,11,19,2,26,11,

%U 23,16,15,2,30,2,25,26,16,14,36,2,22,13,27,2,32,2,21,28

%N a(n) = 1 + Sum_{d|n, d < n} a(d - 1).

%H Seiichi Manyama, <a href="/A343371/b343371.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^2 * A(x^2) + x^3 * A(x^3) + x^4 * A(x^4) + ...

%p a:= proc(n) option remember;

%p 1+add(a(d-1), d=numtheory[divisors](n) minus {n})

%p end:

%p seq(a(n), n=0..75); # _Alois P. Heinz_, Apr 12 2021

%t a[n_] := a[n] = 1 + Sum[If[d < n, a[d - 1], 0], {d, Divisors[n]}]; Table[a[n], {n, 0, 75}]

%t nmax = 75; A[_] = 0; Do[A[x_] = 1/(1 - x) + Sum[x^k A[x^k], {k, 2, nmax}] + O[x]^(nmax + 1) //Normal, nmax + 1]; CoefficientList[A[x], x]

%Y Cf. A003238, A067824, A068336, A167865.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 12 2021