A140670 a(n) = 1 if n is odd; otherwise, a(n) = 2^k - 1 where 2^k is the largest power of 2 that divides n.

1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 31, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 63, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 31, 1, 1, 1, 3, 1, 1

Offset: 1

Author

Michael Somos, May 21 2008

Keywords

• nonn
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A001620, A006519, A135481, A160467, A059841, A330569.

Programs

Mathematica

a[n_] := If[OddQ[n], 1, 2^IntegerExponent[n, 2] - 1]; Array[a, 100] (* Amiram Eldar, Oct 22 2022 *)

PARI

{a(n) = if(n==0, 0, if(n%2, 1, 2^valuation(n, 2) - 1))}

Python

def A140670(n): return max(1,(n&-n)-1) # Chai Wah Wu, Jul 08 2022

Formula

a(n) is multiplicative with a(2^e) = 2^e - 1 if e > 0, a(p^e) = 1 if p > 2.

a(2*n + 1) = 1. a(-n) = a(n). a(2*n) = 2 * a(n) + (-1)^n unless n=0.

Dirichlet g.f.: zeta(s)*(1+2^(1-2s)-2^(1-s))/(1-2^(1-s)). - R. J. Mathar, Feb 07 2011

a(n) = (2*A160467(n))-1. - Antti Karttunen, Nov 18 2017

Sum_{k=1..n} a(k) ~ (1/(2*log(2))) * (n*log(n) + (gamma + log(2)/2 - 1) * n), where gamma is Euler's constant (A001620). - Amiram Eldar, Oct 22 2022

a(n) = A006519(n) - A059841(n). - Ridouane Oudra, Jul 30 2025