A259194 Number of partitions of n into four primes.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 2, 3, 3, 4, 4, 6, 3, 6, 5, 7, 5, 9, 5, 11, 7, 11, 7, 13, 6, 14, 9, 15, 8, 18, 9, 21, 10, 19, 11, 24, 10, 26, 12, 26, 13, 30, 12, 34, 15, 33, 16, 38, 14, 41, 17, 41, 16, 45, 16, 50, 19, 47, 21, 56, 20, 61, 20, 57

Offset: 0

Author

Doug Bell, Jun 20 2015

Keywords

• nonn
• easy

Examples

			a(17) = 3 because 17 can be written as the sum of four primes in exactly three ways: 2+2+2+11, 2+3+5+7 and 2+5+5+5.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 0..5000
• Index entries for sequences related to partitions

Crossrefs

Column k=4 of A117278.

Number of partitions of n into r primes for r = 1..10: A010051, A061358, A068307, this sequence, A259195, A259196, A259197, A259198, A259200, A259201.

Cf. A000040.

Programs

Magma

[0] cat [#RestrictedPartitions(n,4,{d:d in PrimesUpTo(n)}):n in [1..100]]; // Marius A. Burtea, May 07 2019

Mathematica

a[n_] := Length@ IntegerPartitions[n, {4}, Prime@ Range@ PrimePi@ n]; a /@
Range[0, 100] (* Giovanni Resta, Jun 21 2015 *)
Table[Count[IntegerPartitions[n,{4}],?(AllTrue[#,PrimeQ]&)],{n,0,80}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Feb 03 2019 *)

PARI

a(n) = {nb = 0; forpart(p=n, if (#p && (#select(x->isprime(x), Vec(p)) == #p), nb+=1), , [4,4]); nb;} \\ Michel Marcus, Jun 21 2015

Formula

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

a(n) = [x^n y^4] Product_{k>=1} 1/(1 - y*x^prime(k)). - Ilya Gutkovskiy, Apr 18 2019