A340572 Number of partitions of n into 4 parts with at least one prime part.

0, 0, 0, 0, 0, 1, 2, 2, 5, 5, 8, 10, 13, 16, 21, 24, 31, 35, 41, 49, 57, 64, 75, 84, 95, 107, 119, 133, 147, 164, 179, 198, 215, 236, 256, 281, 300, 329, 349, 382, 407, 441, 465, 506, 531, 575, 603, 652, 681, 733, 765, 822, 853, 919, 952, 1019, 1057, 1128, 1166, 1247, 1284

Offset: 0

Author

Wesley Ivan Hurt, Jan 11 2021

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A010051, A026810, A340571.

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, 5)
    end:
a:= n-> coeff(b(n$2, 0), x, 4):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 24 2021

Mathematica

Table[Sum[Sum[Sum[Sign[(PrimePi[k] - PrimePi[k - 1]) + (PrimePi[j] - PrimePi[j - 1]) + (PrimePi[i] - PrimePi[i - 1]) + (PrimePi[n - i - j - k] - PrimePi[n - i - j - k - 1])], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} sign( c(k) + c(j) + c(i) + c(n-i-j-k) ), where c is the prime characteristic (A010051).