A343852 a(n) is the least k > 0 such that the binary expansions of k and of n + k have the same numbers of 0's and of 1's.

5, 10, 9, 20, 21, 18, 17, 40, 19, 42, 38, 36, 37, 34, 33, 80, 35, 38, 37, 84, 35, 76, 74, 72, 73, 74, 70, 68, 69, 66, 65, 160, 67, 70, 69, 76, 67, 74, 73, 168, 75, 70, 69, 152, 67, 148, 146, 144, 71, 146, 145, 148, 140, 140, 138, 136, 137, 138, 134, 132, 133

Offset: 1

Author

Rémy Sigrist, May 03 2021

Keywords

• nonn
• base

Comments

This is the binary analog of A343888.

The required comparisons are (1) that the number of ones in the binary expansion of k must equal the number of ones in the binary expansion of k+n, and (2) that the number of zeroes in the binary expansion of k must equal the number of zeroes in the binary expansion of k+n. See the example below. - Harvey P. Dale, Dec 19 2024

Examples

			The first terms, alongside the binary expansions of a(n) and of n + a(n), are:
  n   a(n)  bin(a(n))  bin(n+a(n))
  --  ----  ---------  -----------
   1     5        101          110
   2    10       1010         1100
   3     9       1001         1100
   4    20      10100        11000
   5    21      10101        11010
   6    18      10010        11000
   7    17      10001        11000
   8    40     101000       110000
   9    19      10011        11100
  10    42     101010       110100
  11    38     100110       110001
  12    36     100100       110000
  13    37     100101       110010
  14    34     100010       110000
  15    33     100001       110000
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..8191

Crossrefs

Cf. A000120, A004757, A023416, A293198, A343888.

Programs

Mathematica

lk[n_]:=Module[{k=1},While[DigitCount[k,2,0]!=DigitCount[n+k,2,0]||DigitCount[k,2,1]!=DigitCount[n+k,2,1],k++];k]; Array[lk,70] (* Harvey P. Dale, Dec 19 2024 *)

PARI

a(n) = { for (k=1, oo, if (#binary(k)==#binary(n+k) && hammingweight(k)==hammingweight(n+k), return (k))) }

Python

def a(n):
  k = 1
  while k.bit_length() != (n+k).bit_length() or bin(k).count('1') != bin(n+k).count('1'): k += 1
  return k
print([a(n) for n in range(1, 62)]) # Michael S. Branicky, May 04 2021

Formula

a(n) <= A004757(n).