A341979 Number of partitions of n into 8 distinct primes (counting 1 as a prime).

1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 4, 0, 3, 0, 3, 1, 7, 0, 6, 1, 6, 1, 11, 0, 11, 2, 11, 3, 19, 1, 18, 3, 18, 5, 30, 4, 28, 6, 30, 10, 45, 6, 40, 11, 46, 16, 63, 11, 60, 19, 69, 25, 88, 18, 86, 32, 97, 36, 121, 32, 123, 47, 131, 55, 164, 49, 164, 69, 181, 80

Offset: 59

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

• nonn

Crossrefs

Cf. A008578, A036497, A219202, A341951, A341973, A341974, A341975, A341976, A341977, A341978, A341980, A341981.

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-1)))(
     `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=59..130);  # 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 - 1]]][
     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, 59, 130}] (* Jean-François Alcover, Feb 24 2022, after Alois P. Heinz *)