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A162510 Dirichlet inverse of A076479.

%I A162510 #54 Mar 08 2024 07:52:46
%S A162510 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,
%T A162510 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,
%U A162510 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
%N A162510 Dirichlet inverse of A076479.
%C A162510 Apart from signs, this sequence is identical to A162512.
%H A162510 Antti Karttunen, <a href="/A162510/b162510.txt">Table of n, a(n) for n = 1..10000</a>
%H A162510 Rafael Jakimczuk, <a href="http://dx.doi.org/10.13140/RG.2.2.27745.48487">Generalizations of Mertens's Formula and k-Free and s-Full Numbers with Prime Divisors in Arithmetic Progression</a>, ResearchGate, 2024.
%H A162510 Vaclav Kotesovec, <a href="/A162510/a162510.jpg">Graph - the asymptotic ratio (100000 terms)</a>.
%H A162510 Gérard P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#multiplicative">Multiplicative functions</a>.
%F A162510 Multiplicative with a(p^e) = 2^(e-1) for any prime p and any positive exponent e.
%F A162510 a(n) = n/2 when n is a power of 2 (A000079).
%F A162510 a(n) = 1 when n is a squarefree number (A005117).
%F A162510 a(n) = 2^A046660(n) = A061142(n)/A034444(n). - _R. J. Mathar_, Nov 02 2016
%F A162510 a(n) = Sum_{d|n} mu(d) * 2^A001222(n/d). - _Daniel Suteu_, May 21 2020
%F A162510 a(1) = 1; a(n) = -Sum_{d|n, d < n} (-1)^omega(n/d) * a(d). - _Ilya Gutkovskiy_, Mar 10 2021
%F A162510 Dirichlet g.f.: (1/zeta(s)) * Product_{p prime} (1/(1 - 2/p^s)). - _Amiram Eldar_, Sep 16 2023
%F A162510 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
%F A162510 From _Vaclav Kotesovec_, Mar 08 2024: (Start)
%F A162510 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))).
%F A162510 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 A162510 f(1) = Product_{p prime, p>2} (1 + 1/(p*(p-2))) = A167864 = 1.51478012813749125771853381230067247330485921179389884042843306025133959...,
%F A162510 f'(1) = f(1) * Sum_{p prime, p>2} (-2*log(p)/((p-1)*(p-2))) = -2*f(1)*A347195 = -2.6035805486753944250682818932032862770113061830543948257159113584026980...
%F A162510 and gamma is the Euler-Mascheroni constant A001620. (End)
%p A162510 A162510 := proc(n)
%p A162510     local a,f;
%p A162510     a := 1;
%p A162510     for f in ifactors(n)[2] do
%p A162510         a := a*2^(op(2,f)-1) ;
%p A162510     end do:
%p A162510     return a;
%p A162510 end proc: # _R. J. Mathar_, May 20 2017
%t A162510 a[n_] := 2^(PrimeOmega[n] - PrimeNu[n]); Array[a, 100] (* _Jean-François Alcover_, Apr 24 2017, after _R. J. Mathar_ *)
%o A162510 (PARI) a(n)=my(f=factor(n)[,2]); 2^(vecsum(f)-#f) \\ _Charles R Greathouse IV_, Nov 02 2016
%o A162510 (Python)
%o A162510 from sympy import factorint
%o A162510 from operator import mul
%o A162510 def a(n):
%o A162510     f=factorint(n)
%o A162510     return 1 if n==1 else reduce(mul, [2**(f[i] - 1) for i in f]) # _Indranil Ghosh_, May 20 2017
%Y A162510 Cf. A076479, A162511, A162512.
%Y A162510 Cf. A000079, A001222, A005117, A034444, A046660, A061142.
%Y A162510 Cf. A167864, A347195.
%K A162510 easy,mult,nonn
%O A162510 1,4
%A A162510 _Gerard P. Michon_, Jul 05 2009