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

0,12

Links

Giovanni Resta, Table of n, a(n) for n = 0..5000

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)} 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

Example

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.

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 *)

Prog

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

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.

Sequence in context: A078705 A346018 A050331 * A194344 A128435 A369219

Adjacent sequences: A259191 A259192 A259193 * A259195 A259196 A259197

Keyword

nonn,easy

Author

Doug Bell, Jun 20 2015

Status

approved