A259195 Number of partitions of n into five primes.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 5, 4, 6, 6, 7, 6, 10, 7, 11, 9, 12, 11, 17, 11, 18, 13, 20, 14, 24, 15, 27, 18, 29, 21, 35, 19, 38, 24, 41, 26, 47, 26, 53, 30, 54, 34, 64, 33, 70, 38, 73, 41, 81, 41, 89, 45, 92, 50, 103, 47, 112, 56, 117, 61, 127, 57

Offset

0,14

Links

David A. Corneth, Table of n, a(n) for n = 0..10000 (first 5001 terms from Doug Bell)

Index entries for sequences related to partitions

Sean A. Irvine, Java program (github)

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} c(i) * c(j) * c(k) * c(l) * c(n-i-j-k-l), where c = A010051. - Wesley Ivan Hurt, Apr 17 2019

a(n) = [x^n y^5] Product_{k>=1} 1/(1 - y*x^prime(k)). - Ilya Gutkovskiy, Apr 18 2019

Example

a(17) = 3 because 17 can be written as the sum of five primes in exactly three ways: 2+2+3+3+7, 2+2+3+5+5, and 3+3+3+3+5.

Mathematica

Array[Count[IntegerPartitions[#, {5}], _?(AllTrue[#, PrimeQ] &)] &, 71] (* Michael De Vlieger, Apr 21 2019 *)

Prog

(PARI) a(n) = {nb = 0; forpart(p=n, if (#p && (#select(x->isprime(x), Vec(p)) == #p), nb+=1), , [5, 5]); nb; } \\ Michel Marcus, Jun 21 2015
(Magma) [0] cat [#RestrictedPartitions(n, 5, {p:p in PrimesUpTo(n)}):n in [1..70]]; // Marius A. Burtea, May 09 2019

Crossrefs

Column k=5 of A117278.

Number of partitions of n into r primes for r = 1..10: A010051, A061358, A068307, A259194, this sequence, A259196, A259197, A259198, A259200, A259201.

Cf. A000040.

Sequence in context: A167227 A048280 A024695 * A143997 A160903 A124229

Adjacent sequences: A259192 A259193 A259194 * A259196 A259197 A259198

Keyword

nonn,easy

Author

Doug Bell, Jun 20 2015

Extensions

More terms from David A. Corneth, Sep 06 2020

Status

approved