A353504 Number of integer partitions of n whose product is less than the product of their multiplicities.

0, 0, 1, 1, 1, 2, 2, 4, 4, 6, 6, 8, 9, 11, 12, 14, 14, 18, 21, 23, 26, 29, 29, 33, 36, 39, 40, 43, 44, 50, 53, 55, 59, 65, 69, 72, 78, 79, 81, 85, 92, 95, 97, 100, 103, 108, 109, 112, 118, 124, 129, 137, 139, 142, 149, 155, 159, 165, 166, 173, 178, 181, 187

Offset: 0

Author

Gus Wiseman, May 19 2022

Keywords

• nonn

Examples

			The a(2) = 1 through a(9) = 6 partitions:
  11   111   1111   2111    21111    22111     221111     222111
                    11111   111111   31111     311111     411111
                                     211111    2111111    2211111
                                     1111111   11111111   3111111
                                                          21111111
                                                          111111111
		

Crossrefs

LHS (product of parts) is counted by A339095, ranked by A003963.

RHS (product of multiplicities) is counted by A266477, ranked by A005361.

The version for greater instead of less is A353505.

The version for equal instead of less is A353506, ranked by A353503.

A181819 gives prime shadow, with an inverse A181821.

A353398 counts partitions with the same product of multiplicities as of shadows, ranked by A353399.

Cf. A002033, A008284, A085629, A097318, A098859, A114640, A116608, A118914, A124010, A304678, A353394, A353500, A353507.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Times@@#