A329632 - OEIS

%I #7 Nov 19 2019 16:36:20

%S 1,1,1,1,2,1,3,1,3,2,4,1,5,1,6,4,6,1,9,2,10,6,13,3,15,6,18,8,22,9,29,

%T 10,30,20,40,22,48,24,57,36,68

%N Number of connected integer partitions of n whose distinct parts are pairwise indivisible.

%C Given an integer partition y of length k, let G(y) be the simple labeled graph with vertices {1..k} and edges between any two vertices i, j such that GCD(y_i, y_j) > 1. For example, G(6,14,15,35) is a 4-cycle. A partition y is said to be connected if G(y) is a connected graph.

%e The a(n) partitions for n = 1, 4, 6, 10, 12, 14:

%e   (1)  (4)    (6)      (10)         (12)           (14)

%e        (2,2)  (3,3)    (5,5)        (6,6)          (7,7)

%e               (2,2,2)  (6,4)        (4,4,4)        (8,6)

%e                        (2,2,2,2,2)  (3,3,3,3)      (10,4)

%e                                     (2,2,2,2,2,2)  (6,4,4)

%e                                                    (2,2,2,2,2,2,2)

%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];

%t zsm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],GCD@@s[[#]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];

%t Table[Length[Select[IntegerPartitions[n],stableQ[#,Divisible]&&Length[zsm[#]]<=1&]],{n,0,30}]

%Y The Heinz numbers of these partitions are given by A329559.

%Y The strict version is A304717.

%Y Connected partitions are A218970.

%Y Pairwise indivisible partitions are A305148.

%Y Cf. A048143, A286518, A286520, A304714, A304716, A305078.

%K nonn,more

%O 0,5

%A _Gus Wiseman_, Nov 18 2019