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

1,2

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.

Links

Table of n, a(n) for n=1..62.

Example

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}

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&]

Crossrefs

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

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

Sequence in context: A325460 A002269 A327445 * A047487 A327203 A048461

Adjacent sequences: A325116 A325117 A325118 * A325120 A325121 A325122

Keyword

nonn

Author

Gus Wiseman, Mar 28 2019

Status

approved