A360070 Numbers for which there exists an integer partition such that the parts have the same mean as the multiplicities.

1, 4, 8, 9, 12, 16, 18, 20, 25, 27, 32, 36, 45, 48, 49, 50, 54, 63, 64, 72, 75, 80, 81, 90, 96, 98, 99, 100, 108, 112, 117, 121, 125, 128, 144, 147, 150, 160, 162, 169, 175, 176, 180, 192, 196, 200, 208, 216, 224, 225, 240, 242, 243, 245, 250, 252, 256, 272

Offset: 1

Author

Gus Wiseman, Jan 27 2023

Keywords

• nonn

Comments

Conjecture: No term > 1 is squarefree.

Examples

			A partition of 20 with the same mean as its multiplicities is (5,4,3,2,1,1,1,1,1,1), so 20 is in the sequence.
		

Crossrefs

Positions of positive terms in A360068, ranked by A359903.

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).

Cf. A005117, A067340, A067538, A240219, A316313, A349156, A359904, A359905, A360069.

Programs

Mathematica

Select[Range[30],Select[IntegerPartitions[#],Mean[#]==Mean[Length/@Split[#]]&]!={}&]

Extensions

a(22)-a(58) from Alois P. Heinz, Jan 29 2023