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A319810
Number of fully periodic integer partitions of n.
5
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.
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 28 2018
STATUS
approved