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A342241
a(n) is the least k > 0 such that the first k bits and the last k bits in the binary expansion of n are the same.
2
1, 1, 2, 1, 3, 1, 3, 1, 4, 1, 2, 1, 4, 1, 4, 1, 5, 1, 2, 1, 5, 1, 2, 1, 5, 1, 5, 1, 5, 1, 5, 1, 6, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 6, 1, 6, 1, 3, 1, 6, 1, 6, 1, 6, 1, 6, 1, 7, 1, 2, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 2, 1, 7, 1, 2, 1, 7, 1, 2
OFFSET
0,3
COMMENTS
This sequence gives the length of the least nonempty prefix that is also a suffix of the binary expansion of a number.
FORMULA
a(n) = 1 iff n = 0 or n is odd.
a(n) <= A070939(n) with equality iff n belongs to A091065.
a(n) = A070939(A342242(n)).
EXAMPLE
For n = 42:
- the binary representation of 42 is "101010",
- the first bit ("1") and the last bit ("0") do not match,
- the first 2 bits ("10") and the last 2 bits ("10") match,
- so a(42) = 2.
PROG
(PARI) a(n) = { my (b=if (n, binary(n), [0])); for (w=1, oo, if (b[1..w]==b[#b+1-w..#b], return (w))) }
(Python)
def a(n):
b = bin(n)[2:]
for i in range(1, len(b)+1):
if b[:i] == b[-i:]: return i
print([a(n) for n in range(87)]) # Michael S. Branicky, Mar 07 2021
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Rémy Sigrist, Mar 07 2021
STATUS
approved