A341978 Number of partitions of n into 7 distinct primes (counting 1 as a prime).

1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 2, 0, 4, 0, 2, 1, 6, 1, 7, 0, 5, 2, 8, 1, 11, 1, 10, 4, 15, 3, 18, 3, 17, 7, 22, 6, 28, 6, 25, 11, 35, 11, 40, 11, 38, 19, 50, 18, 56, 18, 54, 30, 70, 28, 74, 30, 78, 45, 92, 40, 100, 46, 104, 63, 123, 60, 133, 69, 140, 88, 157, 86, 173

Offset: 42

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 42..10000

Crossrefs

Cf. A008578, A036497, A219201, A341950, A341973, A341974, A341975, A341976, A341977, A341979, 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, 8)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 7):
seq(a(n), n=42..114);  # 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, 8}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 7];
Table[a[n], {n, 42, 114}] (* Jean-François Alcover, Feb 24 2022, after Alois P. Heinz *)