A259445 - OEIS

%I #40 Nov 28 2022 08:28:04

%S 1,2,3,2,5,6,7,2,9,10,11,6,13,14,15,2,17,18,19,10,21,22,23,6,25,26,27,

%T 14,29,30,31,2,33,34,35,18,37,38,39,10,41,42,43,22,45,46,47,6,49,50,

%U 51,26,53,54,55,14,57,58,59,30,61,62,63,2,65,66,67,34

%N Multiplicative with a(n) = n if n is odd and a(2^s)=2.

%C If n = 2^s*m with m odd and s > 0 then a(n) = 2*m.

%H G. C. Greubel, <a href="/A259445/b259445.txt">Table of n, a(n) for n = 1..10000</a>

%H Peter Bala, <a href="/A067856/a067856_1.pdf">A signed Dirichlet product of arithmetical functions</a>

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%F From _Peter Bala_, Feb 21 2019: (Start)

%F a(n) = n*gcd(n,2)/gcd(n,2^n).

%F a(2*n) = 2*A000265(2*n); a(2*n+1) = A000265(2*n+1).

%F 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.

%F 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)).

%F (End)

%F From _Peter Bala_, Mar 09 2019: (Start)

%F 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.

%F O.g.f.: Sum_{n >= 1} phi(n)*x^n/(1 + (-x)^n). (End)

%F From _Amiram Eldar_, Nov 28 2022: (Start)

%F Dirichlet g.f.: zeta(s-1)*(1 + 1/2^(s-1) - 2/(2^s-1)).

%F Sum_{k=1..n} a(k) ~ (5/12) * n^2. (End)

%t G[n_] := If[Mod[n, 2] == 0, n/2^(FactorInteger[n][[1, 2]] - 1), n]; Table[G[n], {n, 1, 70}]

%o (PARI) a(n)=n>>max(valuation(n,2)-1,0) \\ _Charles R Greathouse IV_, Jun 28 2015

%Y Cf. A000265, A022998, A018819, A046897, A321558.

%K nonn,mult,easy

%O 1,2

%A _José María Grau Ribas_, Jun 27 2015