A325118 - OEIS

%I #6 Jul 27 2019 14:57:51

%S 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 37,39,40,41,43,44,45,46,47,49,50,51,53,55,59,60,61,62,64,65,67,68,71,

%U 73,75,77,79,80,81,82,83,85,87,88,89,90,91,92,93,94,97,100

%N Heinz numbers of binary carry-connected integer partitions.

%C 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 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 The sequence of terms together with their prime indices begins:

%e    1: {}

%e    2: {1}

%e    3: {2}

%e    4: {1,1}

%e    5: {3}

%e    7: {4}

%e    8: {1,1,1}

%e    9: {2,2}

%e   10: {1,3}

%e   11: {5}

%e   13: {6}

%e   15: {2,3}

%e   16: {1,1,1,1}

%e   17: {7}

%e   19: {8}

%e   20: {1,1,3}

%e   22: {1,5}

%e   23: {9}

%e   25: {3,3}

%e   27: {2,2,2}

%e   29: {10}

%t binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t 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 Select[Range[100],Length[csm[binpos/@PrimePi/@First/@FactorInteger[#]]]<=1&]

%Y Cf. A019565, A048143, A056239, A112798, A247935, A304716, A305078.

%Y Cf. A325098, A325099, A325105, A325109, A325110, A325119.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 28 2019