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%I #21 Aug 27 2023 04:34:56
%S 1,3,2,8,4,6,6,20,6,12,10,16,12,18,8,48,16,18,18,32,12,30,22,40,20,36,
%T 18,48,28,24,30,112,20,48,24,48,36,54,24,80,40,36,42,80,24,66,46,96,
%U 42,60,32,96,52,54,40,120,36,84,58,64,60,90,36,256,48,60,66,128,44,72,70
%N G.f.: x/(1-x) = Sum_{n>=1} a(n)*log(1+x^n)/n.
%C Dirichlet inverse of A117212. - _R. J. Mathar_, Jul 15 2010
%H Paul D. Hanna, <a href="/A162728/b162728.txt">Table of n, a(n) for n = 1..10000</a>
%F a(2n-1) = phi(2n-1); a(2n) = phi(2n)*A090739(n), where A090739(n) = exponent of 2 in 3^(2n)-1.
%F Inverse Mobius transform of A091512, where A091512(n) = exponent of 2 in (2n)^n.
%F Multiplicative: a(m,n) = a(m)*a(n) when gcd(m,n)=1, with a(p) = p-1 for odd prime p and a(2)=3.
%F G.f.: x/(1-x)^2 = Sum_{n>=1} a(n)*x^n/(1+x^n). - _Paul D. Hanna_, Jul 12 2009
%F Dirichlet g.f.: zeta(s-1)/( zeta(s)*(1-2^(1-s)) ). - _R. J. Mathar_, Apr 14 2011
%F a((2*n-1)*2^p) = (p+2)*2^(p-1)* phi(2*n-1), p >= 0. Observe that a(2^p) = A001792(p). - _Johannes W. Meijer_, Jan 26 2013
%F Sum_{k=1..n} a(k) ~ 6*n^2 / Pi^2. - _Vaclav Kotesovec_, Feb 07 2019
%F Multiplicative with a(2^e) = (e+2)*2^(e-1) and a(p^e) = (p-1)*p^(e-1) for an odd prime p. - _Amiram Eldar_, Aug 27 2023
%e x/(1-x) = log(1+x) + 3*log(1+x^2)/2 + 2*log(1+x^3)/3 + 8*log(1+x^4)/4 + 4*log(1+x^5)/5 + 6*log(1+x^6)/6 + 6*log(1+x^7)/7 + 20*log(1+x^8)/8 +...
%p nmax:=71: with(numtheory): for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := (p+2)*2^(p-1)*phi(2*n-1) od: od: seq(a(n), n=1..nmax); # _Johannes W. Meijer_, Jan 26 2013
%t f[p_, e_] := (p-1)*p^(e-1); f[2, e_] := (e+2)*2^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Aug 27 2023 *)
%o (PARI) /* As the inverse Mobius transform of A091512: */
%o {a(n)=sumdiv(n,d,moebius(n/d)*valuation((2*d)^d,2))}
%o (PARI) /* From a(2n-1)=phi(2n-1); a(2n)=phi(2n)*A090739(n), we get: */
%o {a(n)=if(n%2==1,eulerphi(n),eulerphi(n)*valuation(3^n-1,2))}
%o (PARI) /* From x/(1-x) = Sum_{n>=1} a(n)*log(1+x^n)/n, we get: */
%o {a(n)=local(A=[1]);for(k=1,n,A=concat(A,0);A[ #A]=#A*(1-polcoeff(sum(m=1,#A,A[m]/m*log(1+x^m +x*O(x^#A)) ),#A)));A[n]}
%Y Cf. A090739, A091512, A000010 (Euler phi), A220466.
%K mult,nonn,easy
%O 1,2
%A _Paul D. Hanna_, Jul 12 2009
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