%I #9 Nov 17 2018 20:03:04
%S 1,1,1,2,1,3,1,3,2,3,1,6,1,3,3,5,1,7,1,7,3,3,1,13,2,3,4,9,1,13,1,11,3,
%T 3,3,23,1,3,3,20,1,17,1,16,9,3,1,38,2,9,3,23,1,25,3,36,3,3,1,71,1,3,
%U 11,49,3,31,1,52,3,19
%N Number of superperiodic integer partitions of n.
%C An integer partition is superperiodic if either it consists of a single part equal to 1 or its parts have a common divisor > 1 and its multiset of multiplicities is itself superperiodic. For example, (8,8,6,6,4,4,4,4,2,2,2,2) has multiplicities (4,4,2,2) with multiplicities (2,2) with multiplicities (2) with multiplicities (1). The first four of these partitions are periodic and the last is (1), so (8,8,6,6,4,4,4,4,2,2,2,2) is superperiodic.
%e The a(24) = 11 superperiodic partitions:
%e (24)
%e (12,12)
%e (8,8,8)
%e (9,9,3,3)
%e (8,8,4,4)
%e (6,6,6,6)
%e (10,10,2,2)
%e (6,6,6,2,2,2)
%e (6,6,4,4,2,2)
%e (4,4,4,4,4,4)
%e (4,4,4,4,2,2,2,2)
%e (3,3,3,3,3,3,3,3)
%e (2,2,2,2,2,2,2,2,2,2,2,2)
%t wotperQ[m_]:=Or[m=={1},And[GCD@@m>1,wotperQ[Sort[Length/@Split[Sort[m]]]]]];
%t Table[Length[Select[IntegerPartitions[n],wotperQ]],{n,30}]
%Y Cf. A000041, A000837, A018783, A024994, A047966, A098859, A100953, A181819, A182857, A304660, A305563, A317245, A317256, A319151, A319152, A319157.
%K nonn
%O 1,4
%A _Gus Wiseman_, Sep 12 2018