cp's OEIS Frontend

A304716 Number of integer partitions of n whose distinct parts are connected.

%I A304716 #11 Dec 05 2018 05:14:17
%S A304716 1,2,2,3,2,5,2,6,4,9,3,15,4,18,12,25,11,41,17,54,36,72,44,113,69,145,
%T A304716 113,204,153,302,220,394,343,541,475,771,662,1023,968,1398,1314,1929,
%U A304716 1822,2566,2565,3440,3446,4677,4688,6187,6407,8216,8544,10975,11436
%N A304716 Number of integer partitions of n whose distinct parts are connected.
%C A304716 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.
%F A304716 For n > 1, a(n) = A218970(n) + 1. - _Gus Wiseman_, Dec 04 2018
%e A304716 The a(12) = 15 connected integer partitions and their corresponding connected multiset multisystems (see A112798, A302242) are the following.
%e A304716                      (12): {{1,1,2}}
%e A304716                     (6 6): {{1,2},{1,2}}
%e A304716                     (8 4): {{1,1,1},{1,1}}
%e A304716                     (9 3): {{2,2},{2}}
%e A304716                    (10 2): {{1,3},{1}}
%e A304716                   (4 4 4): {{1,1},{1,1},{1,1}}
%e A304716                   (6 3 3): {{1,2},{2},{2}}
%e A304716                   (6 4 2): {{1,2},{1,1},{1}}
%e A304716                   (8 2 2): {{1,1,1},{1},{1}}
%e A304716                 (3 3 3 3): {{2},{2},{2},{2}}
%e A304716                 (4 4 2 2): {{1,1},{1,1},{1},{1}}
%e A304716                 (6 2 2 2): {{1,2},{1},{1},{1}}
%e A304716               (4 2 2 2 2): {{1,1},{1},{1},{1},{1}}
%e A304716             (2 2 2 2 2 2): {{1},{1},{1},{1},{1},{1}}
%e A304716 (1 1 1 1 1 1 1 1 1 1 1 1): {{},{},{},{},{},{},{},{},{},{},{},{}}
%t A304716 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 A304716 Table[Length[Select[IntegerPartitions[n],Length[zsm[Union[#]]]===1&]],{n,30}]
%Y A304716 Cf. A000009, A003963, A048143, A054921, A218970, A285572, A286518, A302242, A304714, A305078, A305079, A322306, A322307.
%K A304716 nonn
%O A304716 1,2
%A A304716 _Gus Wiseman_, May 17 2018
%E A304716 Name changed to distinguish from A218970 by _Gus Wiseman_, Dec 04 2018