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A319810 Number of fully periodic integer partitions of n. 3
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 (list; graph; refs; listen; history; text; internal format)
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: A238791 A007012 A298423 * A325250 A062830 A322366

Adjacent sequences:  A319807 A319808 A319809 * A319811 A319812 A319813

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 28 2018

STATUS

approved

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Last modified April 19 10:35 EDT 2019. Contains 322252 sequences. (Running on oeis4.)