A328871 - OEIS

%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