A348588 Number of partitions of n into 3 parts with at least 1 prime part.

0, 0, 0, 0, 1, 2, 2, 4, 4, 6, 7, 8, 10, 12, 13, 17, 17, 19, 22, 25, 26, 31, 31, 36, 37, 42, 43, 49, 49, 56, 56, 64, 63, 72, 70, 80, 79, 87, 87, 99, 94, 107, 105, 116, 114, 126, 123, 139, 134, 148, 145, 158, 155, 173, 166, 184, 178, 195, 189, 211, 202, 222, 215, 236, 226, 250, 242

Offset: 0

Author

Wesley Ivan Hurt, Oct 24 2021

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A010051, A068307, A235945, A340572.

Programs

Maple

b:= proc(n, i, t) option remember; series(
     `if`(n=0, t, `if`(i<1, 0, expand(x*b(n-i, min(n-i, i),
     `if`(isprime(i), 1, t)))+b(n, i-1, t))), x, 4)
    end:
a:= n-> coeff(b(n$2, 0), x, 3):
seq(a(n), n=0..66);  # Alois P. Heinz, Oct 24 2021

Mathematica

Table[Length@Select[IntegerPartitions[k,{3}],Or@@PrimeQ@#&],{k,0,66}] (* Giorgos Kalogeropoulos, Oct 24 2021 *)

Formula

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} sign(c(i) + c(j) + c(n-i-j)), where c is the prime characteristic (A010051).