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A319179 Number of integer partitions of n that are relatively prime but not aperiodic. Number of integer partitions of n that are aperiodic but not relatively prime. 2

%I #4 Sep 15 2018 15:48:07

%S 0,1,1,1,1,2,1,3,2,6,1,9,1,14,7,17,1,32,1,36,15,55,1,77,6,100,27,121,

%T 1,200,1,209,56,296,19,403,1,489,101,596,1,885,1,947,192,1254,1,1673,

%U 14,1979,297,2336,1,3300,60,3594,490,4564,1,5988,1,6841,800

%N Number of integer partitions of n that are relatively prime but not aperiodic. Number of integer partitions of n that are aperiodic but not relatively prime.

%C An integer partition is aperiodic if its multiplicities are relatively prime.

%e The a(12) = 9 integer partitions that are relatively prime but not aperiodic:

%e (5511),

%e (332211), (333111), (441111),

%e (22221111), (33111111),

%e (222111111),

%e (2211111111),

%e (111111111111).

%e The a(12) = 9 integer partitions that are aperiodic but not relatively prime:

%e (12),

%e (8,4), (9,3), (10,2),

%e (6,3,3), (6,4,2), (8,2,2),

%e (6,2,2,2),

%e (4,2,2,2,2).

%t Table[Length[Select[IntegerPartitions[n],And[GCD@@#==1,GCD@@Length/@Split[#]>1]&]],{n,30}]

%Y Cf. A000837, A018783, A047966, A100953, A305563, A319149, A319160, A319162, A319164.

%K nonn

%O 1,6

%A _Gus Wiseman_, Sep 12 2018

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