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

0,7

Links

Table of n, a(n) for n=0..60.

Index entries for sequences related to partitions

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).

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}]

Crossrefs

Cf. A010051, A026810, A340571.

Sequence in context: A340223 A073707 A238945 * A091609 A183563 A222706

Adjacent sequences: A340569 A340570 A340571 * A340573 A340574 A340575

Keyword

nonn

Author

Wesley Ivan Hurt, Jan 11 2021

Status

approved