A373399 For any number m, let m* be the bi-infinite string obtained by repetition of the binary expansion of m; a(n) is the least k such that the binary expansion of n appears in k*.

1, 2, 1, 4, 2, 5, 1, 8, 4, 2, 5, 9, 5, 11, 1, 16, 8, 4, 9, 18, 2, 5, 11, 17, 9, 21, 5, 19, 11, 23, 1, 32, 16, 8, 17, 4, 18, 9, 19, 34, 18, 2, 21, 37, 5, 11, 23, 33, 17, 37, 9, 38, 21, 5, 11, 35, 19, 43, 11, 39, 23, 47, 1, 64, 32, 16, 33, 8, 34, 17, 35, 68, 4

Offset

1,2

Links

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

Rémy Sigrist, PARI program

Index entries for sequences related to binary expansion of n

Formula

a(n) <= n with equality iff n is a power of 2.

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

Example

The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   1     1       1          1
   2     2      10         10
   3     1      11          1
   4     4     100        100
   5     2     101         10
   6     5     110        101
   7     1     111          1
   8     8    1000       1000
   9     4    1001        100
  10     2    1010         10
  11     5    1011        101
  12     9    1100       1001
  13     5    1101        101
  14    11    1110       1011
  15     1    1111          1
  16    16   10000      10000

Prog

(PARI) \\ See Links section.
(Python)
def a(n):
    target = bin(n)[2:]
    for m in range(1, n):
        b = bin(m)[2:]
        mstar = b*(2*len(target)//len(b))
        if target in mstar:
            return m
    return n
print([a(n) for n in range(1, 74)]) # Michael S. Branicky, Jun 14 2024

Crossrefs

Cf. A326774, A361942, A373553.

Sequence in context: A276133 A307602 A054269 * A086450 A370153 A270439

Adjacent sequences: A373396 A373397 A373398 * A373400 A373401 A373402

Keyword

nonn,base

Author

Rémy Sigrist, Jun 04 2024

Status

approved