A348363 The 1's in the binary expansion of a(n) encode the distances between the 1's in the binary expansion of n.

0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 15, 3, 15, 7, 15, 1, 17, 9, 27, 5, 21, 15, 31, 3, 27, 15, 31, 7, 31, 15, 31, 1, 33, 17, 51, 9, 45, 27, 63, 5, 45, 21, 63, 15, 47, 31, 63, 3, 51, 27, 59, 15, 63, 31, 63, 7, 63, 31, 63, 15, 63, 31, 63, 1, 65, 33, 99, 17, 85, 51

Offset

0,4

Comments

The bit 2^d is set in a(n) iff for some e >= 0, the bits 2^e and 2^(e+d) are set in n.

This sequence has similarities with A067398; here we take the absolute differences, there the sums, of indices of 1's in binary expansions.

All terms are odd, except a(0) = 0.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Rémy Sigrist, Colored scatterplot of the first 2^20 terms (where the color is function of the 2-adic valuation of n, upper red pixels correspond to odd n's)

Index entries for sequences related to binary expansion of n

Formula

a(2*n) = a(n).

a(n) = n iff n = 0 or n belongs to A064896.

a(n) = 1 iff n is a power of 2 (A000079).

a(n) = 3 iff n belongs to A007283.

a(n) = 5 iff n belongs to A020714.

a(n) AND n = n for any odd number n (where AND denotes the bitwise AND operator).

Example

The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     1      10          1
   3     3      11         11
   4     1     100          1
   5     5     101        101
   6     3     110         11
   7     7     111        111
   8     1    1000          1
   9     9    1001       1001
  10     5    1010        101
  11    15    1011       1111
  12     3    1100         11
  13    15    1101       1111
  14     7    1110        111
  15    15    1111       1111

Mathematica

{0}~Join~Array[Total[2^Append[Union@ Abs[Subtract @@@ Permutations[1 + Length[#] - Position[#, 1][[All, 1]] &@ IntegerDigits[#, 2], {2}]], 0]] &, 70] (* Michael De Vlieger, Oct 16 2021 *)

Prog

(PARI) a(n) = { my (b=vector(hammingweight(n))); for (k=1, #b, n-=2^b[k]=valuation(n, 2); ); my (p=setbinop((i, j)->abs(i-j), b)); sum (k=1, #p, 2^p[k]) }
(Python)
def a(n):
    locs = [e for e in range(n.bit_length()) if 1 & (n>>e)]
    diffs = set(abs(e1-e2) for i, e1 in enumerate(locs) for e2 in locs[i:])
    return sum(2**d for d in diffs)
print([a(n) for n in range(71)]) # Michael S. Branicky, Oct 16 2021

Crossrefs

Cf. A000079, A007283, A020714, A064896, A067398.

Sequence in context: A093474 A030101 A162742 * A081432 A318060 A336652

Adjacent sequences: A348360 A348361 A348362 * A348364 A348365 A348366

Keyword

nonn,base

Author

Rémy Sigrist, Oct 15 2021

Status

approved