A360068 Number of integer partitions of n such that the parts have the same mean as the multiplicities.

1, 1, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 6, 0, 0, 0, 6, 0, 7, 0, 1, 0, 0, 0, 0, 90, 0, 63, 0, 0, 0, 0, 11, 0, 0, 0, 436, 0, 0, 0, 0, 0, 0, 0, 0, 2157, 0, 0, 240, 1595, 22, 0, 0, 0, 6464, 0, 0, 0, 0, 0, 0, 0, 0, 11628, 4361, 0, 0, 0, 0, 0, 0, 0, 12927, 0, 0, 621, 0

Offset: 0

Author

Gus Wiseman, Jan 27 2023

Keywords

• nonn

Comments

Note that such a partition cannot be strict for n > 1.

Conjecture: If n is squarefree, then a(n) = 0.

Examples

			The n = 1, 4, 8, 9, 12, 16, 18 partitions (D=13):
  (1)  (22)  (3311)  (333)  (322221)  (4444)      (444222)
             (5111)         (332211)  (43222111)  (444411)
                            (422211)  (43321111)  (552222)
                            (522111)  (53221111)  (555111)
                            (531111)  (54211111)  (771111)
                            (621111)  (63211111)  (822222)
                                                  (D11111)
For example, the partition (4,3,3,3,3,3,2,2,1,1) has mean 5/2, and its multiplicities (1,5,2,2) also have mean 5/2, so it is counted under a(20).
		

Crossrefs

These partitions are ranked by A359903, for prime factors A359904.

Positions of positive terms are A360070.

A000041 counts partitions, strict A000009.

A058398 counts partitions by mean, see also A008284, A327482.

A088529/A088530 gives mean of prime signature (A124010).

A326567/A326568 gives mean of prime indices (A112798).

A360069 counts partitions whose multiplicities have integer mean.

Cf. A067340, A067538, A082550, A240219, A316313, A327475, A349156, A359893, A359897, A359905.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Mean[#]==Mean[Length/@Split[#]]&]],{n,0,30}]