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A326305 Dirichlet g.f.: zeta(s-1) * (1 - 2^(-s)) / zeta(s).

%I A326305 #21 Sep 08 2022 08:46:24
%S A326305 1,0,2,1,4,0,6,2,6,0,10,2,12,0,8,4,16,0,18,4,12,0,22,4,20,0,18,6,28,0,
%T A326305 30,8,20,0,24,6,36,0,24,8,40,0,42,10,24,0,46,8,42,0,32,12,52,0,40,12,
%U A326305 36,0,58,8,60,0,36,16,48,0,66,16,44,0,70,12,72,0,40
%N A326305 Dirichlet g.f.: zeta(s-1) * (1 - 2^(-s)) / zeta(s).
%C A326305 Moebius transform of A026741.
%C A326305 Dirichlet convolution of A002131 with Dirichlet inverse of A000005.
%C A326305 Dirichlet convolution of A000027 with Dirichlet inverse of A001511.
%H A326305 Amiram Eldar, <a href="/A326305/b326305.txt">Table of n, a(n) for n = 1..10000</a>
%F A326305 a(n) = phi(n) if n odd, phi(n) - phi(n/2) if n even, where phi = A000010.
%F A326305 a(n) = Sum_{d|n} mu(n/d) * A026741(d).
%F A326305 a(n) = Sum_{d|n} A007427(n/d) * A002131(d).
%F A326305 a(n) = Sum_{d|n} A092673(n/d) * d.
%F A326305 a(p) = p - 1, where p is odd prime.
%F A326305 Product_{n>=1} 1 / (1 - x^n)^a(n) = g.f. for A299069.
%F A326305 Sum_{k=1..n} a(k) ~ 9*n^2 / (4*Pi^2). - _Vaclav Kotesovec_, Oct 26 2019
%F A326305 Multiplicative with a(2^e) = 0 if e = 1 and 2^(e-2) otherwise, and a(p^e) = (p-1)*p^(e-1) for odd primes p. - _Amiram Eldar_, Nov 30 2020
%t A326305 Table[Sum[MoebiusMu[n/d] Numerator[d/2], {d, Divisors[n]}], {n, 1, 75}]
%t A326305 a[n_] := If[OddQ[n], EulerPhi[n], EulerPhi[n] - EulerPhi[n/2]]; Table[a[n], {n, 1, 75}]
%t A326305 f[2, e_] := If[e == 1, 0, 2^(e - 2)]; f[p_, e_] := (p - 1)*p^(e - 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Nov 30 2020 *)
%o A326305 (Magma) [IsOdd(n) select EulerPhi(n) else EulerPhi(n)-EulerPhi(n div 2) : n in [1..80]]; // _Marius A. Burtea_, Oct 17 2019
%Y A326305 Cf. A000005, A000010, A000027, A001511, A002131, A007427, A008683, A016825 (positions of 0's), A026741, A092673, A299069.
%K A326305 nonn,mult
%O A326305 1,3
%A A326305 _Ilya Gutkovskiy_, Oct 17 2019