OFFSET
1,2
COMMENTS
If n = 2^s*m with m odd and s > 0 then a(n) = 2*m.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
FORMULA
From Peter Bala, Feb 21 2019: (Start)
a(n) = n*gcd(n,2)/gcd(n,2^n).
O.g.f.: x*(1 + 4*x + x^2)/(1 - x^2)^2 - 2*( F(x^2) + F(x^4) + F(x^8) + ... ), where F(x) = x/(1 - x)^2.
O.g.f. for reciprocals: Sum_{n >= 1} (1/a(n))*x^n = (3/4)*L(x) - (1/4)*L(-x) + (1/4)*( L(x^2) + L(x^4) + L(x^8) + ... ), where L(x) = log(1/(1 - x)).
(End)
From Peter Bala, Mar 09 2019: (Start)
a(n) = (-1)^(n+1)*Sum_ {d divides n} (-1)^(d+n/d)*phi(d), where phi(n) = A000010(n) is the Euler totient function. Cf. the identity n = Sum_ {d divides n} phi(d). Cf. A046897 and A321558.
O.g.f.: Sum_{n >= 1} phi(n)*x^n/(1 + (-x)^n). (End)
From Amiram Eldar, Nov 28 2022: (Start)
Dirichlet g.f.: zeta(s-1)*(1 + 1/2^(s-1) - 2/(2^s-1)).
Sum_{k=1..n} a(k) ~ (5/12) * n^2. (End)
MATHEMATICA
G[n_] := If[Mod[n, 2] == 0, n/2^(FactorInteger[n][[1, 2]] - 1), n]; Table[G[n], {n, 1, 70}]
PROG
(PARI) a(n)=n>>max(valuation(n, 2)-1, 0) \\ Charles R Greathouse IV, Jun 28 2015
CROSSREFS
KEYWORD
nonn,mult,easy
AUTHOR
José María Grau Ribas, Jun 27 2015
STATUS
approved