cp's OEIS Frontend

A319149 Number of superperiodic integer partitions of n.

%I A319149 #9 Nov 17 2018 20:03:04
%S A319149 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 A319149 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 A319149 11,49,3,31,1,52,3,19
%N A319149 Number of superperiodic integer partitions of n.
%C A319149 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 A319149 The a(24) = 11 superperiodic partitions:
%e A319149   (24)
%e A319149   (12,12)
%e A319149   (8,8,8)
%e A319149   (9,9,3,3)
%e A319149   (8,8,4,4)
%e A319149   (6,6,6,6)
%e A319149   (10,10,2,2)
%e A319149   (6,6,6,2,2,2)
%e A319149   (6,6,4,4,2,2)
%e A319149   (4,4,4,4,4,4)
%e A319149   (4,4,4,4,2,2,2,2)
%e A319149   (3,3,3,3,3,3,3,3)
%e A319149   (2,2,2,2,2,2,2,2,2,2,2,2)
%t A319149 wotperQ[m_]:=Or[m=={1},And[GCD@@m>1,wotperQ[Sort[Length/@Split[Sort[m]]]]]];
%t A319149 Table[Length[Select[IntegerPartitions[n],wotperQ]],{n,30}]
%Y A319149 Cf. A000041, A000837, A018783, A024994, A047966, A098859, A100953, A181819, A182857, A304660, A305563, A317245, A317256, A319151, A319152, A319157.
%K A319149 nonn
%O A319149 1,4
%A A319149 _Gus Wiseman_, Sep 12 2018