A341951 Number of partitions of n into 8 primes (counting 1 as a prime).

1, 1, 2, 2, 4, 4, 7, 7, 11, 10, 15, 14, 21, 19, 27, 23, 35, 30, 44, 37, 54, 44, 67, 55, 81, 65, 96, 75, 115, 89, 133, 102, 155, 116, 180, 134, 206, 153, 236, 171, 271, 194, 305, 220, 346, 242, 391, 273, 438, 305, 489, 334, 551, 374, 608, 412, 674, 447, 750, 494, 823

Offset: 8

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

• nonn

Crossrefs

Cf. A008578, A034891, A259198, A341945, A341946, A341947, A341948, A341949, A341950.

Programs

Maple

b:= proc(n, i) option remember; series(`if`(n=0, 1,
     `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i)))(
     `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 9)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 8):
seq(a(n), n=8..68);  # Alois P. Heinz, Feb 24 2021

Mathematica

b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
     If[i < 0, 0, Function[p, If[p > n, 0, x*b[n - p, i]]][
     If[i == 0, 1, Prime[i]]] + b[n, i - 1]]], {x, 0, 9}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 8];
Table[a[n], {n, 8, 68}] (* Jean-François Alcover, Feb 15 2022, after Alois P. Heinz *)