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

1, 2, 3, 5, 7, 10, 11, 13, 15, 17, 19, 22, 23, 29, 30, 31, 34, 37, 39, 41, 43, 46, 47, 51, 53, 55, 59, 61, 62, 65, 67, 71, 73, 77, 79, 82, 83, 85, 87, 89, 91, 93, 94, 97, 101, 102, 103, 107, 109, 110, 113, 115, 118, 119, 127, 129, 130, 131, 134, 137, 139, 141

Offset: 1

Author

Gus Wiseman, Mar 28 2019

Keywords

• nonn

Comments

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.

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.

Examples

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  10: {1,3}
  11: {5}
  13: {6}
  15: {2,3}
  17: {7}
  19: {8}
  22: {1,5}
  23: {9}
  29: {10}
  30: {1,2,3}
  31: {11}
  34: {1,7}
  37: {12}
  39: {2,6}
  41: {13}
  43: {14}
		

Crossrefs

Cf. A019565, A048143, A056239, A112798, A304714, A304716, A305078.

Cf. A325098, A325099, A325100, A325105, A325110, A325118.

Programs

Mathematica

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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]]]]]]]]];
Select[Range[100],SquareFreeQ[#]&&Length[csm[binpos/@PrimePi/@First/@FactorInteger[#]]]<=1&]