A319149 Number of superperiodic integer partitions of n.

1, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 6, 1, 3, 3, 5, 1, 7, 1, 7, 3, 3, 1, 13, 2, 3, 4, 9, 1, 13, 1, 11, 3, 3, 3, 23, 1, 3, 3, 20, 1, 17, 1, 16, 9, 3, 1, 38, 2, 9, 3, 23, 1, 25, 3, 36, 3, 3, 1, 71, 1, 3, 11, 49, 3, 31, 1, 52, 3, 19

Offset: 1

Author

Gus Wiseman, Sep 12 2018

Keywords

• nonn

Comments

An integer partition is superperiodic if either it consists of a single part equal to 1 or its parts have a common divisor > 1 and its multiset of multiplicities is itself superperiodic. For example, (8,8,6,6,4,4,4,4,2,2,2,2) has multiplicities (4,4,2,2) with multiplicities (2,2) with multiplicities (2) with multiplicities (1). The first four of these partitions are periodic and the last is (1), so (8,8,6,6,4,4,4,4,2,2,2,2) is superperiodic.

Examples

			The a(24) = 11 superperiodic partitions:
  (24)
  (12,12)
  (8,8,8)
  (9,9,3,3)
  (8,8,4,4)
  (6,6,6,6)
  (10,10,2,2)
  (6,6,6,2,2,2)
  (6,6,4,4,2,2)
  (4,4,4,4,4,4)
  (4,4,4,4,2,2,2,2)
  (3,3,3,3,3,3,3,3)
  (2,2,2,2,2,2,2,2,2,2,2,2)
		

Crossrefs

Cf. A000041, A000837, A018783, A024994, A047966, A098859, A100953, A181819, A182857, A304660, A305563, A317245, A317256, A319151, A319152, A319157.

Programs

Mathematica

wotperQ[m_]:=Or[m=={1},And[GCD@@m>1,wotperQ[Sort[Length/@Split[Sort[m]]]]]];
Table[Length[Select[IntegerPartitions[n],wotperQ]],{n,30}]