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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 12 2018
STATUS
approved