A331383 Number of integer partitions of n whose sum of primes of parts is equal to their product of parts.

0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 1, 1, 1, 2, 2, 2, 1, 4, 2, 2, 2, 4, 2, 3, 4, 1, 3, 4, 5, 0, 3, 3, 1, 6, 2, 1, 5, 4, 2, 3, 4, 2, 2, 3, 1, 5, 2, 3, 4, 6, 5, 2, 7, 1, 3, 5, 3, 4, 2, 5, 5, 4, 7, 3, 6, 4, 4, 2, 4, 4, 3, 9, 4, 3, 5, 3, 5, 4, 4, 4, 3, 7, 4, 2, 8, 2, 3

Offset: 1

Author

Gus Wiseman, Jan 16 2020

Keywords

• nonn

Examples

			The a(n) partitions for n = 7, 9, 18, 24:
  (4,3)  (6,3)    (12,4,1,1)                 (19,4,1)
         (4,4,1)  (11,4,1,1,1)               (18,4,1,1)
                  (8,5,1,1,1,1,1)            (9,6,1,1,1,1,1,1,1,1,1)
                  (4,2,2,2,1,1,1,1,1,1,1,1)
For example, (4,4,1) has sum of primes of parts 7+7+2 = 16 and product of parts 4*4*1 = 16, so is counted under a(9).
		

Crossrefs

The Heinz numbers of these partitions are given by A331384.

Numbers divisible by the sum of their prime factors are A036844.

Partitions whose product is divisible by their sum are A057568.

Numbers divisible by the sum of their prime indices are A324851.

Product of prime indices is divisible by sum of prime indices: A326149.

Partitions whose Heinz number is divisible by their sum are A330950.

Partitions whose Heinz number is divisible by their sum of primes: A330953.

Sum of prime factors is divisible by sum of prime indices: A331380

Partitions whose product divides their sum of primes are A331381.

Cf. A000040, A001414, A324850, A330954, A331378, A331379, A331382, A331415, A331416.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Times@@#==Plus@@Prime/@#&]],{n,30}]

PARI

a(n) = my(c=0); forpart(v=n, if(vecprod(Vec(v))==sum(i=1, #v, prime(v[i])), c++)); c; \\ Jinyuan Wang, Feb 14 2025

Extensions

a(71)-a(87) from Robert Price, Apr 10 2020