A356352 a(n) = GCD of run lengths in binary expansion of n.

0, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Author

Rémy Sigrist, Oct 15 2022

Keywords

• nonn
• base

Comments

a(0) = 0 as the GCD of an empty list (we consider here that the binary expansion of 0 has no runs).

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..16384
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A001196, A005811, A101211, A284559 (LCM).

Programs

Mathematica

{0}~Join~Array[GCD @@ Map[Length, Split@ IntegerDigits[#, 2]] &, 104] (* Michael De Vlieger, Oct 17 2022 *)

PARI

a(n) = { my (r=[]); while (n, my (v=valuation(n+n%2, 2)); n\=2^v; r=concat(v, r)); gcd(r) }

Python

from math import gcd
from itertools import groupby
def a(n):
    if n == 0: return 0 # by convention
    return gcd(*(len(list(g)) for k, g in groupby(bin(n)[2:])))
print([a(n) for n in range(87)]) # Michael S. Branicky, Oct 15 2022

Formula

a(A001196(n)) = 2*a(n).

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