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A325119 Heinz numbers of binary carry-connected strict integer partitions.

%I A325119 #5 Jul 27 2019 14:57:51
%S A325119 1,2,3,5,7,10,11,13,15,17,19,22,23,29,30,31,34,37,39,41,43,46,47,51,
%T A325119 53,55,59,61,62,65,67,71,73,77,79,82,83,85,87,89,91,93,94,97,101,102,
%U A325119 103,107,109,110,113,115,118,119,127,129,130,131,134,137,139,141
%N A325119 Heinz numbers of binary carry-connected strict integer partitions.
%C A325119 A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. An integer partition is binary carry-connected if the graph whose vertices are the parts and whose edges are binary carries is connected.
%C A325119 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are squarefree numbers whose prime indices are binary carry-connected. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%e A325119 The sequence of terms together with their prime indices begins:
%e A325119    1: {}
%e A325119    2: {1}
%e A325119    3: {2}
%e A325119    5: {3}
%e A325119    7: {4}
%e A325119   10: {1,3}
%e A325119   11: {5}
%e A325119   13: {6}
%e A325119   15: {2,3}
%e A325119   17: {7}
%e A325119   19: {8}
%e A325119   22: {1,5}
%e A325119   23: {9}
%e A325119   29: {10}
%e A325119   30: {1,2,3}
%e A325119   31: {11}
%e A325119   34: {1,7}
%e A325119   37: {12}
%e A325119   39: {2,6}
%e A325119   41: {13}
%e A325119   43: {14}
%t A325119 binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A325119 csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A325119 Select[Range[100],SquareFreeQ[#]&&Length[csm[binpos/@PrimePi/@First/@FactorInteger[#]]]<=1&]
%Y A325119 Cf. A019565, A048143, A056239, A112798, A304714, A304716, A305078.
%Y A325119 Cf. A325098, A325099, A325100, A325105, A325110, A325118.
%K A325119 nonn
%O A325119 1,2
%A A325119 _Gus Wiseman_, Mar 28 2019