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%I #6 Nov 12 2019 19:23:29
%S 1,1,2,2,3,2,4,2,4,3,5,2,6,2,7,5,7,2,10,2,11,7,14,2,16,4,19,8,22,2,30,
%T 3,29,14,37,8,48,4,50,19,59,5,82,4,81,28,93,8,128,9,128,38,147,8,199,
%U 19,196,52,223,12,308
%N Number of integer partitions of n whose distinct parts are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).
%C A partition with no two distinct parts divisible is said to be stable, and a partition with no two distinct parts relatively prime is said to be intersecting, so these are just stable intersecting partitions.
%e The a(1) = 1 through a(10) = 5 partitions (A = 10):
%e 1 2 3 4 5 6 7 8 9 A
%e 11 111 22 11111 33 1111111 44 333 55
%e 1111 222 2222 111111111 64
%e 111111 11111111 22222
%e 1111111111
%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t Table[Length[Select[IntegerPartitions[n],stableQ[Union[#],Divisible]&&stableQ[Union[#],GCD[#1,#2]==1&]&]],{n,0,30}]
%Y The Heinz numbers of these partitions are A329366.
%Y Replacing "intersecting" with "relatively prime" gives A328676.
%Y Stable partitions are A305148.
%Y Intersecting partitions are A328673.
%Y Cf. A000837, A285573, A303362, A305148, A316476, A328671, A328677.
%K nonn
%O 0,3
%A _Gus Wiseman_, Nov 12 2019
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