A304714 - OEIS

%I #8 Nov 30 2019 16:59:10

%S 1,1,1,1,1,2,1,2,2,3,2,5,2,5,5,6,5,10,6,12,12,13,14,21,17,23,26,30,31,

%T 46,38,51,55,61,70,87,85,102,116,128,138,171,169,204,225,245,272,319,

%U 334,383,429,464,515,593,629,715,790,861,950,1082

%N Number of connected strict integer partitions of n.

%C Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.

%e The a(19) = 6 strict integer partitions are (19), (9,6,4), (10,5,4), (10,6,3), (12,4,3), (8,6,3,2). Taking the normalized prime factors of each part (see A112798, A302242), we have the following connected multiset multisystems.

%e        (19): {{8}}

%e     (9,6,4): {{2,2},{1,2},{1,1}}

%e    (10,5,4): {{1,3},{3},{1,1}}

%e    (10,6,3): {{1,3},{1,2},{2}}

%e    (12,4,3): {{1,1,2},{1,1},{2}}

%e   (8,6,3,2): {{1,1,1},{1,2},{2},{1}}

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

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[zsm[#]]===1&]],{n,60}]

%Y The Heinz numbers of these partitions are given by A328513.

%Y Cf. A000009, A003963, A048143, A054921, A076078, A259936, A281116, A285572, A285573, A286518, A286520, A302242.

%K nonn

%O 1,6

%A _Gus Wiseman_, May 17 2018