A356365 For any nonnegative integer n with binary expansion Sum_{k = 1..w} 2^e_k, let m be the least integer such that the values e_k mod m are all distinct; a(n) = Sum_{k = 1..w} 2^(e_k mod m).

0, 1, 1, 3, 1, 5, 3, 7, 1, 3, 3, 11, 3, 13, 7, 15, 1, 3, 3, 19, 6, 7, 7, 23, 3, 25, 11, 27, 7, 29, 15, 31, 1, 3, 6, 7, 3, 7, 7, 39, 5, 11, 7, 43, 14, 15, 15, 47, 3, 7, 19, 51, 7, 53, 23, 55, 7, 57, 27, 59, 15, 61, 31, 63, 1, 5, 3, 7, 5, 7, 7, 71, 3, 13, 14, 15

Offset

0,4

Comments

See A293390 for the corresponding m's.

Links

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

Index entries for sequences related to binary expansion of n

Formula

A000120(a(n)) = A000120(n).

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

a(2^k - 1) = 2^k - 1 for any k >= 0.

Example

The first terms, alongside their binary expansions and the corresponding m's, are:
  n    a(n)  bin(n)   bin(a(n))  m
  ---  ----  -------  ---------  -
    0     0        0          0  0
    1     1        1          1  1
    2     1       10          1  1
    3     3       11         11  2
    4     1      100          1  1
    5     5      101        101  3
    6     3      110         11  2
    7     7      111        111  3
    8     1     1000          1  1
    9     3     1001         11  2
   10     3     1010         11  3
   11    11     1011       1011  4
   12     3     1100         11  2
   13    13     1101       1101  4
   14     7     1110        111  3
   15    15     1111       1111  4
   16     1    10000          1  1

Prog

(PARI) a(n) = { my (b=vector(hammingweight(n))); for (i=1, #b, n-=2^b[i]=valuation(n, 2); ); for (m=1, oo, if (#Set(b%m)==#b, b%=m; break; ); ); sum(i=1, #b, 2^b[i]); }

Crossrefs

Cf. A000120, A064895, A293390.

Sequence in context: A334212 A082725 A308655 * A217663 A078701 A299766

Adjacent sequences: A356362 A356363 A356364 * A356366 A356367 A356368

Keyword

nonn,base

Author

Rémy Sigrist, Oct 16 2022

Status

approved