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A325118 Heinz numbers of binary carry-connected integer partitions.

%I A325118 #6 Jul 27 2019 14:57:51
%S A325118 1,2,3,4,5,7,8,9,10,11,13,15,16,17,19,20,22,23,25,27,29,30,31,32,34,
%T A325118 37,39,40,41,43,44,45,46,47,49,50,51,53,55,59,60,61,62,64,65,67,68,71,
%U A325118 73,75,77,79,80,81,82,83,85,87,88,89,90,91,92,93,94,97,100
%N A325118 Heinz numbers of binary carry-connected integer partitions.
%C A325118 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 A325118 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are 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 A325118 The sequence of terms together with their prime indices begins:
%e A325118    1: {}
%e A325118    2: {1}
%e A325118    3: {2}
%e A325118    4: {1,1}
%e A325118    5: {3}
%e A325118    7: {4}
%e A325118    8: {1,1,1}
%e A325118    9: {2,2}
%e A325118   10: {1,3}
%e A325118   11: {5}
%e A325118   13: {6}
%e A325118   15: {2,3}
%e A325118   16: {1,1,1,1}
%e A325118   17: {7}
%e A325118   19: {8}
%e A325118   20: {1,1,3}
%e A325118   22: {1,5}
%e A325118   23: {9}
%e A325118   25: {3,3}
%e A325118   27: {2,2,2}
%e A325118   29: {10}
%t A325118 binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A325118 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 A325118 Select[Range[100],Length[csm[binpos/@PrimePi/@First/@FactorInteger[#]]]<=1&]
%Y A325118 Cf. A019565, A048143, A056239, A112798, A247935, A304716, A305078.
%Y A325118 Cf. A325098, A325099, A325105, A325109, A325110, A325119.
%K A325118 nonn
%O A325118 1,2
%A A325118 _Gus Wiseman_, Mar 28 2019