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

1,4

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.

Links

Table of n, a(n) for n=1..70.

Example

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)

Mathematica

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

Crossrefs

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

Sequence in context: A364818 A378181 A353338 * A344462 A321887 A046051

Adjacent sequences: A319146 A319147 A319148 * A319150 A319151 A319152

Keyword

nonn

Author

Gus Wiseman, Sep 12 2018

Status

approved