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

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, 14, 29, 30, 31, 2, 33, 34, 35, 18, 37, 38, 39, 10, 41, 42, 43, 22, 45, 46, 47, 6, 49, 50, 51, 26, 53, 54, 55, 14, 57, 58, 59, 30, 61, 62, 63, 2, 65, 66, 67, 34

Offset: 1

Author

José María Grau Ribas, Jun 27 2015

Keywords

• nonn
• mult
• easy

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
• Peter Bala, A signed Dirichlet product of arithmetical functions
• Index to divisibility sequences

Crossrefs

Cf. A000265, A022998, A018819, A046897, A160467, A321558.

Programs

Maple

A259445 := proc(n::integer)
    local a, pe, p,e ;
    a := 1 ;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if p = 2 then
            a := 2*a ;
        else
            a := a*p^e  ;
        end if;
    end do:
    a;
end proc:
seq(A259445(n),n=1..80) ; # R. J. Mathar, Feb 21 2025

Mathematica

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

PARI

a(n)=n>>max(valuation(n,2)-1,0) \\ Charles R Greathouse IV, Jun 28 2015

Formula

From Peter Bala, Feb 21 2019: (Start)

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

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

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)

a(n) = n /A160467(n). - R. J. Mathar, Feb 21 2025