cp's OEIS Frontend

A304717 Number of connected strict integer partitions of n with pairwise indivisible parts.

%I A304717 #6 May 17 2018 21:37:28
%S A304717 1,1,1,1,1,1,1,1,1,2,1,1,1,3,2,2,1,3,2,4,3,5,2,5,4,6,3,7,6,9,5,9,8,13,
%T A304717 10,15,9,15,13,18,14,22,21,26,19,29,24,36,31,40,35,45,38,54,55,59,55,
%U A304717 70,69,84,74,89,86,107,103,119,115,143,143,159
%N A304717 Number of connected strict integer partitions of n with pairwise indivisible parts.
%C A304717 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 A304717 The a(34) = 13 connected strict integer partitions with pairwise indivisible parts are (34), (18,16), (20,14), (22,12), (24,10), (26,8), (28,6), (30,4), (14,12,8), (15,10,9), (20,8,6), (14,10,6,4), (15,9,6,4). Their corresponding multiset multisystems (see A112798, A302242) are the following.
%e A304717          (34): {{1,7}}
%e A304717        (30 4): {{1,2,3},{1,1}}
%e A304717        (28 6): {{1,1,4},{1,2}}
%e A304717        (26 8): {{1,6},{1,1,1}}
%e A304717       (24 10): {{1,1,1,2},{1,3}}
%e A304717       (22 12): {{1,5},{1,1,2}}
%e A304717       (20 14): {{1,1,3},{1,4}}
%e A304717      (20 8 6): {{1,1,3},{1,1,1},{1,2}}
%e A304717       (18 16): {{1,2,2},{1,1,1,1}}
%e A304717     (15 10 9): {{2,3},{1,3},{2,2}}
%e A304717    (15 9 6 4): {{2,3},{2,2},{1,2},{1,1}}
%e A304717     (14 12 8): {{1,4},{1,1,2},{1,1,1}}
%e A304717   (14 10 6 4): {{1,4},{1,3},{1,2},{1,1}}
%t A304717 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 A304717 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[zsm[#]]===1&&Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]==={}&]],{n,30}]
%Y A304717 Cf. A000009, A003963, A006126, A048143, A051424, A054921, A076078, A259936, A281116, A285572, A285573, A286518, A286520, A293993, A302242, A303362, A304714, A304716.
%K A304717 nonn
%O A304717 1,10
%A A304717 _Gus Wiseman_, May 17 2018