login
Number of partitions of n into 8 prime powers (including 1).
9

%I #10 Feb 22 2022 03:35:53

%S 1,1,2,3,5,6,10,13,19,23,32,38,51,60,77,90,113,128,158,179,215,240,

%T 287,316,373,409,475,517,599,645,741,799,908,971,1104,1173,1326,1408,

%U 1580,1670,1874,1967,2198,2310,2563,2680,2976,3097,3426,3566,3926,4070,4485

%N Number of partitions of n into 8 prime powers (including 1).

%p q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:

%p b:= proc(n, i, t) option remember; `if`(n=0,

%p `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+

%p `if`(q(i), b(n-i, min(n-i, i), t-1), 0)))

%p end:

%p a:= n-> b(n$2, 8):

%p seq(a(n), n=8..60); # _Alois P. Heinz_, Feb 05 2021

%t q[n_] := q[n] = Length[FactorInteger[n]] < 2;

%t b[n_, i_, t_] := b[n, i, t] = If[n == 0,

%t If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +

%t If[q[i], b[n - i, Min[n - i, i], t - 1], 0]]];

%t a[n_] := b[n, n, 8];

%t Table[a[n], {n, 8, 60}] (* _Jean-François Alcover_, Feb 22 2022, after _Alois P. Heinz *)

%Y Cf. A000961, A010055, A071330, A341112, A341122, A341123, A341124, A341125, A341127.

%K nonn

%O 8,3

%A _Ilya Gutkovskiy_, Feb 05 2021