A364525 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.

0, 0, 1, 1, 2, 5, 9, 18, 36, 73, 145, 290, 580, 1159, 2319, 4637, 9273, 18544, 37083, 74157, 148330, 296658, 593311, 1186613, 2373208, 4746380, 9492687, 18985447, 37970821, 75941497, 151882704, 303764828, 607528497, 1215054675, 2430104713, 4860217541

Offset: 1

Author

Robert P. P. McKone, Dec 22 2023

Keywords

• nonn

Crossrefs

Cf. A000720.

Cf. A166444.

Cf. A365062 (sum of pi(x) + 1 for n>0).

Programs

Mathematica

p[n_] := p[n] = PrimePi[n];
pv[n_] := pv[n] = p[n]*(p[n] + 1)/2;
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];
a[n_] := a[n] = Module[{c = 0, k = 1}, While[k <= n, c += v[n, k]; k++]; c]; Table[a[n], {n, 1, 36}]