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A259445 Multiplicative with a(n) = n if n is odd and a(2^s)=2.

%I A259445 #46 Feb 21 2025 06:01:41
%S A259445 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 A259445 14,29,30,31,2,33,34,35,18,37,38,39,10,41,42,43,22,45,46,47,6,49,50,
%U A259445 51,26,53,54,55,14,57,58,59,30,61,62,63,2,65,66,67,34
%N A259445 Multiplicative with a(n) = n if n is odd and a(2^s)=2.
%C A259445 If n = 2^s*m with m odd and s > 0 then a(n) = 2*m.
%H A259445 G. C. Greubel, <a href="/A259445/b259445.txt">Table of n, a(n) for n = 1..10000</a>
%H A259445 Peter Bala, <a href="/A067856/a067856_1.pdf">A signed Dirichlet product of arithmetical functions</a>
%H A259445 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F A259445 From _Peter Bala_, Feb 21 2019: (Start)
%F A259445 a(n) = n*gcd(n,2)/gcd(n,2^n).
%F A259445 a(2*n) = 2*A000265(2*n); a(2*n+1) = A000265(2*n+1).
%F A259445 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 A259445 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 A259445 (End)
%F A259445 From _Peter Bala_, Mar 09 2019: (Start)
%F A259445 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 A259445 O.g.f.: Sum_{n >= 1} phi(n)*x^n/(1 + (-x)^n). (End)
%F A259445 From _Amiram Eldar_, Nov 28 2022: (Start)
%F A259445 Dirichlet g.f.: zeta(s-1)*(1 + 1/2^(s-1) - 2/(2^s-1)).
%F A259445 Sum_{k=1..n} a(k) ~ (5/12) * n^2. (End)
%F A259445 a(n) = n /A160467(n). - _R. J. Mathar_, Feb 21 2025
%p A259445 A259445 := proc(n::integer)
%p A259445     local a, pe, p,e ;
%p A259445     a := 1 ;
%p A259445     for pe in ifactors(n)[2] do
%p A259445         p := op(1,pe) ;
%p A259445         e := op(2,pe) ;
%p A259445         if p = 2 then
%p A259445             a := 2*a ;
%p A259445         else
%p A259445             a := a*p^e  ;
%p A259445         end if;
%p A259445     end do:
%p A259445     a;
%p A259445 end proc:
%p A259445 seq(A259445(n),n=1..80) ; # _R. J. Mathar_, Feb 21 2025
%t A259445 G[n_] := If[Mod[n, 2] == 0, n/2^(FactorInteger[n][[1, 2]] - 1), n]; Table[G[n], {n, 1, 70}]
%o A259445 (PARI) a(n)=n>>max(valuation(n,2)-1,0) \\ _Charles R Greathouse IV_, Jun 28 2015
%Y A259445 Cf. A000265, A022998, A018819, A046897, A160467, A321558.
%K A259445 nonn,mult,easy
%O A259445 1,2
%A A259445 _José María Grau Ribas_, Jun 27 2015