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%I #35 Jan 22 2024 06:26:42
%S 0,0,1,1,2,5,9,18,36,73,145,290,580,1159,2319,4637,9273,18544,37083,
%T 74157,148330,296658,593311,1186613,2373208,4746380,9492687,18985447,
%U 37970821,75941497,151882704,303764828,607528497,1215054675,2430104713,4860217541
%N a(n) is the number of distinct ways to partition the set {1,2,...,n} into nonempty subsets such that the sum of the pi(x)*(pi(x) + 1)/2 values of each subset's size x equals n, where pi() is the prime counting function given by A000720.
%t p[n_] := p[n] = PrimePi[n];
%t pv[n_] := pv[n] = p[n]*(p[n] + 1)/2;
%t v[n_, k_] := v[n, k] = Module[{c = 0, i = 1}, If[k == 1, Return[If[pv[n] == n, 1, 0]]]; While[i < n - k + 2, If[pv[i] <= n, c += v[n - i, k - 1]]; i++]; c];
%t a[n_] := a[n] = Module[{c = 0, k = 1}, While[k <= n, c += v[n, k]; k++]; c]; Table[a[n], {n, 1, 36}]
%Y Cf. A000720.
%Y Cf. A166444.
%Y Cf. A365062 (sum of pi(x) + 1 for n>0).
%K nonn
%O 1,5
%A _Robert P. P. McKone_, Dec 22 2023