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A319164
Number of integer partitions of n that are neither relatively prime nor aperiodic.
8
0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 5, 0, 2, 2, 5, 0, 6, 0, 9, 2, 2, 0, 17, 1, 2, 3, 17, 0, 18, 0, 22, 2, 2, 2, 48, 0, 2, 2, 48, 0, 34, 0, 58, 11, 2, 0, 111, 1, 14, 2, 103, 0, 65, 2, 141, 2, 2, 0, 264, 0, 2, 19, 231, 2, 116, 0, 299, 2, 42
OFFSET
1,6
COMMENTS
A partition is aperiodic if its multiplicities are relatively prime.
EXAMPLE
The a(24) = 17 integer partitions:
(12,12),
(8,8,8),
(6,6,6,6), (8,8,4,4), (9,9,3,3), (10,10,2,2),
(4,4,4,4,4,4), (6,6,3,3,3,3), (6,6,4,4,2,2), (6,6,6,2,2,2), (8,8,2,2,2,2),
(3,3,3,3,3,3,3,3), (4,4,4,4,2,2,2,2), (6,6,2,2,2,2,2,2),
(4,4,4,2,2,2,2,2,2),
(4,4,2,2,2,2,2,2,2,2),
(2,2,2,2,2,2,2,2,2,2,2,2).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], And[GCD@@#>1, GCD@@Length/@Split[#]>1]&]], {n, 30}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 12 2018
STATUS
approved