login
A319160
Number of integer partitions of n whose multiplicities appear with relatively prime multiplicities.
6
1, 2, 2, 4, 5, 7, 11, 16, 22, 31, 45, 58, 83, 108, 142, 188, 250, 315, 417, 528, 674, 861, 1094, 1363, 1724, 2152, 2670, 3311, 4105, 5021, 6193, 7561, 9216, 11219, 13614, 16419, 19886, 23920, 28733, 34438, 41272, 49184, 58746, 69823, 82948, 98380, 116567
OFFSET
1,2
COMMENTS
From Gus Wiseman, Jul 11 2023: (Start)
A partition is aperiodic (A000837) if its multiplicities are relatively prime. This sequence counts partitions whose multiplicities are aperiodic.
For example:
- The multiplicities of (5,3) are (1,1), with multiplicities (2), with common divisor 2, so it is not counted under a(8).
- The multiplicities of (3,2,2,1) are (2,1,1), with multiplicities (2,1), which are relatively prime, so it is counted under a(8).
- The multiplicities of (3,3,1,1) are (2,2), with multiplicities (2), with common divisor 2, so it is not counted under a(8).
- The multiplicities of (4,4,4,3,3,3,2,1) are (3,3,1,1), with multiplicities (2,2), which have common divisor 2, so it is not counted under a(24).
(End)
EXAMPLE
The a(8) = 16 partitions:
(8),
(44),
(332), (422), (611),
(2222), (3221), (4211), (5111),
(22211), (32111), (41111),
(221111), (311111),
(2111111),
(11111111).
Missing from this list are: (53), (62), (71), (431), (521), (3311).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], GCD@@Length/@Split[Sort[Length/@Split[#]]]==1&]], {n, 30}]
CROSSREFS
These partitions have ranks A319161.
For distinct instead of relatively prime multiplicities we have A325329.
Sequence in context: A362608 A374688 A325330 * A292382 A296561 A300121
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 12 2018
STATUS
approved