A319810 Number of fully periodic integer partitions of n.

1, 2, 2, 3, 2, 5, 2, 5, 4, 6, 2, 11, 2, 8, 7, 11, 2, 17, 2, 18, 9, 15, 2, 32, 5, 22, 12, 34, 2, 54, 2, 49, 16, 51, 10, 94, 2, 77, 23, 112, 2, 152, 2, 148, 47, 165, 2, 258, 7, 247, 52, 286, 2, 400, 17, 402, 78, 439, 2, 657, 2, 594, 131, 711, 24

Offset

1,2

Comments

An integer partition is fully periodic iff either it is a singleton or it is a periodic partition (meaning its multiplicities have a common divisor > 1) with fully periodic multiplicities.

Links

Table of n, a(n) for n=1..65.

Example

The a(12) = 11 fully periodic integer partitions:
  (12)
  (6,6)
  (4,4,4)
  (5,5,1,1)
  (4,4,2,2)
  (3,3,3,3)
  (3,3,3,1,1,1)
  (3,3,2,2,1,1)
  (2,2,2,2,2,2)
  (2,2,2,2,1,1,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1)
Periodic partitions missing from this list are:
  (4,4,1,1,1,1)
  (3,3,1,1,1,1,1,1)
  (2,2,2,1,1,1,1,1,1)
  (2,2,1,1,1,1,1,1,1,1)
The first non-uniform fully periodic partition is (4,4,3,3,2,2,2,2,1,1,1,1).
The first periodic integer partition that is not fully periodic is (2,2,1,1,1,1).

Mathematica

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

Crossrefs

Cf. A000837, A018783, A047966, A098859, A100953, A305563, A319149, A319160, A319162, A319163, A319164, A319811.

Sequence in context: A007012 A343371 A298423 * A325250 A062830 A322366

Adjacent sequences: A319807 A319808 A319809 * A319811 A319812 A319813

Keyword

nonn

Author

Gus Wiseman, Sep 28 2018

Status

approved