A331381 Number of integer partitions of n whose sum of primes of parts is divisible by their product of parts.

1, 1, 1, 1, 1, 3, 1, 5, 2, 6, 6, 5, 5, 7, 4, 7, 7, 7, 10, 8, 9, 6, 10, 9, 9, 15, 7, 12, 10, 14, 10, 10, 8, 8, 15, 10, 7, 16, 13, 9, 10, 14, 12, 10, 8, 14, 11, 13, 11, 16, 15, 14, 15, 15, 10, 14, 18, 11, 12, 13, 13, 18, 21, 15, 16, 19, 16, 15, 8, 17, 17

Offset: 0

Author

Gus Wiseman, Jan 16 2020

Keywords

• nonn

Examples

			The a(n) partitions for n = 1, 5, 7, 8, 9, 13, 14:
  1  221    43       311111    63         7411           65111
     311    511      11111111  441        721111         322211111
     11111  3211               711        43111111       311111111111
            22111              42111      421111111      11111111111111
            1111111            2211111    3211111111
                               111111111  22111111111
                                          1111111111111
		

Crossrefs

The Heinz numbers of these partitions are given by A331382.

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.

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

Partitions whose product is equal to their sum of primes are A331383.

Product of prime indices equals sum of prime factors: A331384.

Cf. A000040, A001414, A324850, A330953, A330954, A331378, A331379, A331415, A331416.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Divisible[Plus@@Prime/@#,Times@@#]&]],{n,0,30}]