cp's OEIS Frontend

A329632 Number of connected integer partitions of n whose distinct parts are pairwise indivisible.

%I A329632 #7 Nov 19 2019 16:36:20
%S A329632 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 A329632 10,30,20,40,22,48,24,57,36,68
%N A329632 Number of connected integer partitions of n whose distinct parts are pairwise indivisible.
%C A329632 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 A329632 The a(n) partitions for n = 1, 4, 6, 10, 12, 14:
%e A329632   (1)  (4)    (6)      (10)         (12)           (14)
%e A329632        (2,2)  (3,3)    (5,5)        (6,6)          (7,7)
%e A329632               (2,2,2)  (6,4)        (4,4,4)        (8,6)
%e A329632                        (2,2,2,2,2)  (3,3,3,3)      (10,4)
%e A329632                                     (2,2,2,2,2,2)  (6,4,4)
%e A329632                                                    (2,2,2,2,2,2,2)
%t A329632 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A329632 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 A329632 Table[Length[Select[IntegerPartitions[n],stableQ[#,Divisible]&&Length[zsm[#]]<=1&]],{n,0,30}]
%Y A329632 The Heinz numbers of these partitions are given by A329559.
%Y A329632 The strict version is A304717.
%Y A329632 Connected partitions are A218970.
%Y A329632 Pairwise indivisible partitions are A305148.
%Y A329632 Cf. A048143, A286518, A286520, A304714, A304716, A305078.
%K A329632 nonn,more
%O A329632 0,5
%A A329632 _Gus Wiseman_, Nov 18 2019