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

Author

Gerard P. Michon, Jul 05 2009

Keywords

• easy
• mult
• nonn

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.

Crossrefs

Cf. A076479, A162511, A162512.

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

Cf. A167864, A347195.

Programs

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 *)

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

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)