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%I #53 Mar 14 2023 11:38:02
%S 1,2,2,4,2,4,2,8,4,4,2,8,2,4,4,16,2,8,2,8,4,4,2,16,4,4,8,8,2,8,2,32,4,
%T 4,4,16,2,4,4,16,2,8,2,8,8,4,2,32,4,8,4,8,2,16,4,16,4,4,2,16,2,4,8,64,
%U 4,8,2,8,4,8,2,32,2,4,8,8,4,8,2,32,16,4,2,16,4,4,4,16,2,16,4,8,4,4,4
%N Replace each prime factor of n with 2: a(n) = 2^bigomega(n), where bigomega = A001222, number of prime factors counted with multiplicity.
%C The inverse Möbius transform of A162510. - _R. J. Mathar_, Feb 09 2011
%H R. Zumkeller, <a href="/A061142/b061142.txt">Table of n, a(n) for n = 1..10005</a>
%H R. J. Mathar, <a href="http://arxiv.org/abs/1106.4038">Survey of Dirichlet Series of Multiplicative Arithmetic Functions</a>, arXiv:1106.4038 [math.NT], 2011-2012. See eq. (2.12).
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F a(n) = Sum_{d divides n} 2^(bigomega(d)-omega(d)) = Sum_{d divides n} 2^(A001222(d) - A001221(d)). - _Benoit Cloitre_, Apr 30 2002
%F a(n) = A000079(A001222(n)), i.e., a(n)=2^bigomega(n). - _Emeric Deutsch_, Feb 13 2005
%F Totally multiplicative with a(p) = 2. - _Franklin T. Adams-Watters_, Oct 04 2006
%F Dirichlet g.f.: Product_{p prime} 1/(1-2*p^(-s)). - _Ralf Stephan_, Mar 28 2015
%F a(n) = A001316(A156552(n)). - _Antti Karttunen_, May 29 2017
%F Dirichlet g.f.: zeta(s)^2 * Product_{p prime} 1/(1 - 1/(p^s - 1)^2). - _Vaclav Kotesovec_, Mar 14 2023
%e a(100)=16 since 100=2*2*5*5 and so a(100)=2*2*2*2.
%p with(numtheory): seq(2^bigomega(n),n=1..95);
%t Table[2^PrimeOmega[n], {n, 1, 95}] (* _Jean-François Alcover_, Jun 08 2013 *)
%o (PARI) a(n)=direuler(p=1,n,1/(1-2*X))[n] /* _Ralf Stephan_, Mar 28 2015 */
%o (PARI) a(n) = 2^bigomega(n); \\ _Michel Marcus_, Aug 08 2017
%Y Cf. A000079, A001222, A001316, A034444, A069205 (partial sums), A123667, A124508, A156552.
%K easy,nonn,mult
%O 1,2
%A _Henry Bottomley_, May 29 2001
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