A341145 Number of partitions of n into 8 distinct prime powers (including 1).

1, 0, 1, 1, 2, 1, 2, 3, 5, 5, 6, 7, 10, 10, 13, 16, 19, 21, 26, 30, 34, 37, 44, 52, 58, 66, 73, 85, 94, 106, 115, 136, 146, 165, 178, 204, 215, 248, 263, 298, 318, 356, 372, 426, 443, 494, 520, 585, 603, 681, 702, 781, 815, 906, 929, 1044, 1071, 1178, 1223

Offset: 39

Author

Ilya Gutkovskiy, Feb 05 2021

Keywords

• nonn

Crossrefs

Cf. A000961, A010055, A341126, A341132, A341137, A341140, A341141, A341142, A341143, A341144.

Programs

Maple

q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(q(i), b(n-i, min(n-i, i-1), t-1), 0)))
    end:
a:= n-> b(n$2, 8):
seq(a(n), n=39..97);  # Alois P. Heinz, Feb 05 2021

Mathematica

q[n_] := q[n] = Length[FactorInteger[n]] < 2;
b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[q[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 8];
Table[a[n], {n, 39, 97}] (* Jean-François Alcover, Feb 22 2022, after Alois P. Heinz *)