A162510 Dirichlet inverse of A076479.

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 8, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 4, 2, 1, 1, 1, 16, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 8, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 2, 1, 1, 2, 32, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 8, 8, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 16, 1, 2, 2, 4, 1, 1, 1, 4, 1

Offset

1,4

Comments

Apart from signs, this sequence is identical to A162512.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Rafael Jakimczuk, Generalizations of Mertens's Formula and k-Free and s-Full Numbers with Prime Divisors in Arithmetic Progression, ResearchGate, 2024.

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms).

Gérard P. Michon, Multiplicative functions.

Formula

Multiplicative with a(p^e) = 2^(e-1) for any prime p and any positive exponent e.

a(n) = n/2 when n is a power of 2 (A000079).

a(n) = 1 when n is a squarefree number (A005117).

a(n) = 2^A046660(n) = A061142(n)/A034444(n). - R. J. Mathar, Nov 02 2016

a(n) = Sum_{d|n} mu(d) * 2^A001222(n/d). - Daniel Suteu, May 21 2020

a(1) = 1; a(n) = -Sum_{d|n, d < n} (-1)^omega(n/d) * a(d). - Ilya Gutkovskiy, Mar 10 2021

Dirichlet g.f.: (1/zeta(s)) * Product_{p prime} (1/(1 - 2/p^s)). - Amiram Eldar, Sep 16 2023

Sum_{k=1..n} 1/a(k) = c * n + o(n), where c = Product_{p prime} (1 - 1/(p*(2*p-1))) = 0.74030830284678515949... (Jakimczuk, 2024, Theorem 2.4, p. 16). - Amiram Eldar, Mar 08 2024

From Vaclav Kotesovec, Mar 08 2024: (Start)

Dirichlet g.f.: zeta(s) * (1 + 1/(2^s*(2^s - 2))) * f(s), where f(s) = Product_{p prime, p>2} (1 + 1/(p^s*(p^s - 2))).

Sum_{k=1..n} a(k) ~ (f(1)*n / (4*log(2))) * (log(n) - 1 + gamma + 5*log(2)/2 + f'(1)/f(1)), where

f(1) = Product_{p prime, p>2} (1 + 1/(p*(p-2))) = A167864 = 1.51478012813749125771853381230067247330485921179389884042843306025133959...,

f'(1) = f(1) * Sum_{p prime, p>2} (-2*log(p)/((p-1)*(p-2))) = -2*f(1)*A347195 = -2.6035805486753944250682818932032862770113061830543948257159113584026980...

and gamma is the Euler-Mascheroni constant A001620. (End)

Maple

A162510 := proc(n)
    local a, f;
    a := 1;
    for f in ifactors(n)[2] do
        a := a*2^(op(2, f)-1) ;
    end do:
    return a;
end proc: # R. J. Mathar, May 20 2017

Mathematica

a[n_] := 2^(PrimeOmega[n] - PrimeNu[n]); Array[a, 100] (* Jean-François Alcover, Apr 24 2017, after R. J. Mathar *)

Prog

(PARI) a(n)=my(f=factor(n)[, 2]); 2^(vecsum(f)-#f) \\ Charles R Greathouse IV, Nov 02 2016
(Python)
from sympy import factorint
from operator import mul
def a(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [2**(f[i] - 1) for i in f]) # Indranil Ghosh, May 20 2017

Crossrefs

Cf. A076479, A162511, A162512.

Cf. A000079, A001222, A005117, A034444, A046660, A061142.

Cf. A167864, A347195.

Sequence in context: A351655 A351656 A162512 * A292589 A297404 A235388

Adjacent sequences: A162507 A162508 A162509 * A162511 A162512 A162513

Keyword

easy,mult,nonn

Author

Gerard P. Michon, Jul 05 2009

Status

approved