cp's OEIS Frontend

A305504 Heinz numbers of integer partitions whose distinct parts plus 1 are connected.

%I A305504 #6 Jun 04 2018 09:22:19
%S A305504 1,2,3,4,5,7,8,9,10,11,13,16,17,19,20,22,23,25,27,29,31,32,33,34,37,
%T A305504 40,41,43,44,46,47,49,50,53,55,57,59,61,62,64,66,67,68,71,73,79,80,81,
%U A305504 82,83,85,88,89,92,93,94,97,99,100,101,103,107,109,110,113,115
%N A305504 Heinz numbers of integer partitions whose distinct parts plus 1 are connected.
%C A305504 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A305504 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 greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A partition y is said to be connected if G(U(y + 1)) is a connected graph, where U(y + 1) is the set of distinct successors of the parts of y.
%C A305504 This is intended to be a cleaner form of A305078, where the treatment of empty multisets is arbitrary.
%e A305504 The sequence of entries together with the corresponding twice-prime-factored multiset partitions (see A275024) begins:
%e A305504    1: {}
%e A305504    2: {{1}}
%e A305504    3: {{2}}
%e A305504    4: {{1},{1}}
%e A305504    5: {{1,1}}
%e A305504    7: {{3}}
%e A305504    8: {{1},{1},{1}}
%e A305504    9: {{2},{2}}
%e A305504   10: {{1},{1,1}}
%e A305504   11: {{1,2}}
%e A305504   13: {{4}}
%e A305504   16: {{1},{1},{1},{1}}
%e A305504   17: {{1,1,1}}
%e A305504   19: {{2,2}}
%e A305504   20: {{1},{1},{1,1}}
%e A305504   22: {{1},{1,2}}
%t A305504 primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A305504 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 A305504 Select[Range[300],Length[zsm[primeMS[#]+1]]<=1&]
%Y A305504 Cf. A001221, A048143, A056239, A112798, A275024, A286518, A290103, A302242, A304714, A304716, A305052, A305078, A305079, A305501.
%K A305504 nonn
%O A305504 1,2
%A A305504 _Gus Wiseman_, Jun 03 2018