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

1,9

Links

Table of n, a(n) for n=1..87.

Example

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).

Mathematica

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

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.

Sequence in context: A275656 A228825 A324381 * A201208 A006513 A105224

Adjacent sequences: A331380 A331381 A331382 * A331384 A331385 A331386

Keyword

nonn,more

Author

Gus Wiseman, Jan 16 2020

Extensions

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

Status

approved