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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.

%I A140670 #24 Jul 30 2025 10:26:58
%S A140670 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,
%T A140670 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,
%U A140670 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
%N 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.
%H A140670 Antti Karttunen, <a href="/A140670/b140670.txt">Table of n, a(n) for n = 1..16384</a>
%F A140670 a(n) is multiplicative with a(2^e) = 2^e - 1 if e > 0, a(p^e) = 1 if p > 2.
%F A140670 a(2*n + 1) = 1. a(-n) = a(n). a(2*n) = 2 * a(n) + (-1)^n unless n=0.
%F A140670 Dirichlet g.f.: zeta(s)*(1+2^(1-2s)-2^(1-s))/(1-2^(1-s)). - _R. J. Mathar_, Feb 07 2011
%F A140670 a(n) = (2*A160467(n))-1. - _Antti Karttunen_, Nov 18 2017
%F A140670 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
%F A140670 a(n) = A006519(n) - A059841(n). - _Ridouane Oudra_, Jul 30 2025
%t A140670 a[n_] := If[OddQ[n], 1, 2^IntegerExponent[n, 2] - 1]; Array[a, 100] (* _Amiram Eldar_, Oct 22 2022 *)
%o A140670 (PARI) {a(n) = if(n==0, 0, if(n%2, 1, 2^valuation(n, 2) - 1))}
%o A140670 (Python)
%o A140670 def A140670(n): return max(1,(n&-n)-1) # _Chai Wah Wu_, Jul 08 2022
%Y A140670 Cf. A001620, A006519, A135481, A160467, A059841, A330569.
%K A140670 nonn,mult
%O A140670 1,4
%A A140670 _Michael Somos_, May 21 2008