A325097 Heinz numbers of integer partitions whose distinct parts have no binary carries.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 35, 36, 37, 38, 41, 42, 43, 47, 48, 49, 52, 53, 54, 56, 57, 58, 59, 61, 63, 64, 67, 69, 71, 72, 73, 74, 76, 79, 81, 83, 84, 86, 89, 95, 96, 97, 98, 99, 101

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.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose distinct prime indices have no binary carries.

Links

Table of n, a(n) for n=1..67.

Example

Most small numbers are in the sequence, however the sequence of non-terms together with their prime indices begins:
  10: {1,3}
  15: {2,3}
  20: {1,1,3}
  22: {1,5}
  30: {1,2,3}
  34: {1,7}
  39: {2,6}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}
  46: {1,9}
  50: {1,3,3}
  51: {2,7}
  55: {3,5}
  60: {1,1,2,3}
  62: {1,11}
  65: {3,6}
  66: {1,2,5}
  68: {1,1,7}
  70: {1,3,4}

Mathematica

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Select[Range[100], stableQ[PrimePi/@First/@FactorInteger[#], Intersection[binpos[#1], binpos[#2]]!={}&]&]

Crossrefs

Cf. A000110, A000720, A001222, A050315, A056239, A080572, A112798, A247935.

Cf. A325094, A325095, A325096, A325097, A325098, A325100, A325102, A325103.

Sequence in context: A122156 A336547 A371455 * A333126 A330697 A030700

Adjacent sequences: A325094 A325095 A325096 * A325098 A325099 A325100

Keyword

nonn

Author

Gus Wiseman, Mar 27 2019

Status

approved