A308093 For any n > 0, let f_n be the lexicographically earliest sequence of distinct positive terms such that the concatenation of the binary representation of its terms, without leading zeros, corresponds to the binary representation of n repeated indefinitely. Apparently, n always appears in f_n. a(n) gives the index of n in f_n.

1, 1, 2, 1, 3, 3, 3, 1, 3, 2, 8, 3, 8, 5, 4, 1, 3, 3, 8, 3, 5, 8, 15, 3, 8, 8, 15, 5, 10, 7, 5, 1, 3, 3, 8, 2, 7, 7, 15, 3, 7, 3, 9, 7, 7, 15, 28, 3, 8, 7, 15, 7, 15, 7, 24, 5, 10, 9, 17, 7, 11, 9, 6, 1, 3, 3, 8, 3, 7, 7, 15, 3, 5, 8, 14, 8, 14, 11, 28, 3, 7

Offset

1,3

Comments

In other words, f_n(a(n)) = n.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..8192

Rémy Sigrist, PARI program for A308093

Formula

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

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

a(n) = 2 iff n = A007582(k) for some k > 0.

Example

The first terms, alongside the binary representations of n and the first terms of f_n, are:
  n   a(n)  bin(n)  bin(f_n)
  --  ----  ------  -----------------------------------------
   1     1       1  1,...
   2     1      10  10,...
   3     2      11  1,11,...
   4     1     100  100,...
   5     3     101  10,1,101,...
   6     3     110  1,10,110,...
   7     3     111  1,11,111,...
   8     1    1000  1000,...
   9     3    1001  100,1,1001,...
  10     2    1010  10,1010,...
  11     8    1011  10,1,110,11,101,1101,11011,1011,...
  12     3    1100  1,100,1100,...
  13     8    1101  1,10,11,101,110,1110,11101,1101,...
  14     5    1110  1,110,11,10,1110,...
  15     4    1111  1,11,111,1111,...
  16     1   10000  10000,...
  17     3   10001  1000,1,10001,...
  18     3   10010  100,10,10010,...
  19     8   10011  100,1,1100,11,1001,11001,110011,10011,...
  20     3   10100  10,100,10100,...

Prog

(PARI) See Links section.

Crossrefs

Cf. A007582.

Sequence in context: A333224 A124770 A334300 * A264154 A302641 A099246

Adjacent sequences: A308090 A308091 A308092 * A308094 A308095 A308096

Keyword

nonn,base

Author

Rémy Sigrist, May 12 2019

Status

approved