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%I #16 Jul 11 2023 15:56:15
%S 1,2,2,4,5,7,11,16,22,31,45,58,83,108,142,188,250,315,417,528,674,861,
%T 1094,1363,1724,2152,2670,3311,4105,5021,6193,7561,9216,11219,13614,
%U 16419,19886,23920,28733,34438,41272,49184,58746,69823,82948,98380,116567
%N Number of integer partitions of n whose multiplicities appear with relatively prime multiplicities.
%C From _Gus Wiseman_, Jul 11 2023: (Start)
%C A partition is aperiodic (A000837) if its multiplicities are relatively prime. This sequence counts partitions whose multiplicities are aperiodic.
%C For example:
%C - The multiplicities of (5,3) are (1,1), with multiplicities (2), with common divisor 2, so it is not counted under a(8).
%C - 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).
%C - 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).
%C - 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).
%C (End)
%e The a(8) = 16 partitions:
%e (8),
%e (44),
%e (332), (422), (611),
%e (2222), (3221), (4211), (5111),
%e (22211), (32111), (41111),
%e (221111), (311111),
%e (2111111),
%e (11111111).
%e Missing from this list are: (53), (62), (71), (431), (521), (3311).
%t Table[Length[Select[IntegerPartitions[n], GCD@@Length/@Split[Sort[Length/@Split[#]]]==1&]],{n,30}]
%Y These partitions have ranks A319161.
%Y For distinct instead of relatively prime multiplicities we have A325329.
%Y Cf. A000837, A001597, A007916, A047966, A071625, A098859, A100953, A181819, A182850, A182857, A305563, A319149, A319162, A319164.
%K nonn
%O 1,2
%A _Gus Wiseman_, Sep 12 2018
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