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%I #5 Sep 15 2018 15:47:48
%S 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,
%T 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,
%U 264,0,2,19,231,2,116,0,299,2,42
%N Number of integer partitions of n that are neither relatively prime nor aperiodic.
%C A partition is aperiodic if its multiplicities are relatively prime.
%e The a(24) = 17 integer partitions:
%e (12,12),
%e (8,8,8),
%e (6,6,6,6), (8,8,4,4), (9,9,3,3), (10,10,2,2),
%e (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),
%e (3,3,3,3,3,3,3,3), (4,4,4,4,2,2,2,2), (6,6,2,2,2,2,2,2),
%e (4,4,4,2,2,2,2,2,2),
%e (4,4,2,2,2,2,2,2,2,2),
%e (2,2,2,2,2,2,2,2,2,2,2,2).
%t Table[Length[Select[IntegerPartitions[n],And[GCD@@#>1,GCD@@Length/@Split[#]>1]&]],{n,30}]
%Y Cf. A000837, A018783, A047966, A098859, A100953, A305563, A319149, A319160, A319162, A319165.
%K nonn
%O 1,6
%A _Gus Wiseman_, Sep 12 2018