A341974 Number of partitions of n into 3 distinct primes (counting 1 as a prime).

1, 0, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 3, 4, 2, 5, 4, 5, 2, 5, 2, 7, 4, 6, 3, 8, 3, 9, 4, 7, 2, 9, 3, 10, 5, 9, 4, 12, 3, 13, 6, 12, 4, 14, 3, 16, 6, 13, 3, 16, 3, 19, 7, 14, 3, 19, 5, 21, 6, 15, 3, 23, 5, 23, 7, 18, 5, 26, 5, 26, 7, 21, 5, 29, 4, 28, 9, 25, 4, 30, 4

Offset: 6

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 6..10000

Crossrefs

Cf. A008578, A036497, A125688, A341946, A341973, A341975, A341976, A341977.

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, 4)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 3):
seq(a(n), n=6..90);  # 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, 4}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 3];
Table[a[n], {n, 6, 1000}] (* Jean-François Alcover, Jul 13 2021, after Alois P. Heinz *)