A342242 For any n > 0, a(n) is the least positive number whose binary expansion is both a prefix and a suffix of the binary expansion of n; a(0) = 0.

0, 1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 14, 1, 16, 1, 2, 1, 20, 1, 2, 1, 24, 1, 26, 1, 28, 1, 30, 1, 32, 1, 2, 1, 4, 1, 2, 1, 40, 1, 2, 1, 44, 1, 2, 1, 48, 1, 50, 1, 52, 1, 6, 1, 56, 1, 58, 1, 60, 1, 62, 1, 64, 1, 2, 1, 4, 1, 2, 1, 72, 1, 2, 1, 4, 1, 2, 1

Offset: 0

Author

Rémy Sigrist, Mar 07 2021

Keywords

• nonn
• base
• easy

Comments

All terms belong to A091065.

Examples

			For n = 814:
- the binary expansion of 814 is "1100101110",
- "1" does not match "0",
- "11" does not match "10",
- "110" matches "110",
- so the binary representation of a(814) is "110",
- and a(814) = 6.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8192
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A091065, A342241, A342241.

Programs

PARI

a(n) = { my (b=if (n, binary(n), [0])); for (w=1, oo, if (b[1..w]==b[#b+1-w..#b], return (fromdigits(b[1..w],2)))) }

Python

def a(n):
  b = bin(n)[2:]
  for i in range(1, len(b)+1):
    if b[:i] == b[-i:]: return int(b[:i], 2)
print([a(n) for n in range(80)]) # Michael S. Branicky, Mar 07 2021

Formula

a(n) = 1 iff n is an odd number.

a(n) <= n with equality iff n belongs to A091065.

a(n) = n mod 2^A342241(n).

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