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
Examples
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.
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)
Crossrefs
Programs
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Magma
[0] cat [#RestrictedPartitions(n,5,{p:p in PrimesUpTo(n)}):n in [1..70]]; // Marius A. Burtea, May 09 2019 -
Mathematica
Array[Count[IntegerPartitions[#, {5}], ?(AllTrue[#, PrimeQ] &)] &, 71] (* _Michael De Vlieger, Apr 21 2019 *) -
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
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
Extensions
More terms from David A. Corneth, Sep 06 2020