A319160 Number of integer partitions of n whose multiplicities appear with relatively prime multiplicities.

1, 2, 2, 4, 5, 7, 11, 16, 22, 31, 45, 58, 83, 108, 142, 188, 250, 315, 417, 528, 674, 861, 1094, 1363, 1724, 2152, 2670, 3311, 4105, 5021, 6193, 7561, 9216, 11219, 13614, 16419, 19886, 23920, 28733, 34438, 41272, 49184, 58746, 69823, 82948, 98380, 116567

Offset: 1

Author

Gus Wiseman, Sep 12 2018

Keywords

• nonn

Comments

From Gus Wiseman, Jul 11 2023: (Start)

A partition is aperiodic (A000837) if its multiplicities are relatively prime. This sequence counts partitions whose multiplicities are aperiodic.

For example:

- The multiplicities of (5,3) are (1,1), with multiplicities (2), with common divisor 2, so it is not counted under a(8).

- The multiplicities of (3,2,2,1) are (2,1,1), with multiplicities (2,1), which are relatively prime, so it is counted under a(8).

- The multiplicities of (3,3,1,1) are (2,2), with multiplicities (2), with common divisor 2, so it is not counted under a(8).

- The multiplicities of (4,4,4,3,3,3,2,1) are (3,3,1,1), with multiplicities (2,2), which have common divisor 2, so it is not counted under a(24).

(End)

Examples

			The a(8) = 16 partitions:
  (8),
  (44),
  (332), (422), (611),
  (2222), (3221), (4211), (5111),
  (22211), (32111), (41111),
  (221111), (311111),
  (2111111),
  (11111111).
Missing from this list are: (53), (62), (71), (431), (521), (3311).
		

Crossrefs

These partitions have ranks A319161.

For distinct instead of relatively prime multiplicities we have A325329.

Cf. A000837, A001597, A007916, A047966, A071625, A098859, A100953, A181819, A182850, A182857, A305563, A319149, A319162, A319164.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], GCD@@Length/@Split[Sort[Length/@Split[#]]]==1&]],{n,30}]