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A326306 Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 - p^(1 - s) + p^(-s)).

%I A326306 #35 Dec 13 2024 10:12:50
%S A326306 1,2,2,4,2,4,2,8,5,4,2,8,2,4,4,16,2,10,2,8,4,4,2,16,7,4,14,8,2,8,2,32,
%T A326306 4,4,4,20,2,4,4,16,2,8,2,8,10,4,2,32,9,14,4,8,2,28,4,16,4,4,2,16,2,4,
%U A326306 10,64,4,8,2,8,4,8,2,40,2,4,14,8,4,8,2,32,41,4,2,16,4
%N A326306 Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 - p^(1 - s) + p^(-s)).
%C A326306 Inverse Moebius transform of A003557.
%C A326306 Dirichlet convolution of A000203 with A097945.
%H A326306 Vaclav Kotesovec, <a href="/A326306/b326306.txt">Table of n, a(n) for n = 1..10000</a>
%H A326306 Steven R. Finch, <a href="https://arxiv.org/abs/math/0605019">Idempotents and Nilpotents Modulo n</a>, arXiv:math/0605019 [math.NT], 2006-2017.
%F A326306 G.f.: Sum_{k>=1} (k / rad(k)) * x^k / (1 - x^k), where rad = A007947.
%F A326306 a(n) = Sum_{d|n} A003557(d).
%F A326306 a(n) = Sum_{d|n} mu(n/d) * phi(n/d) * sigma(d), where mu = A008683, phi = A000010 and sigma = A000203.
%F A326306 a(p) = 2, where p is prime.
%F A326306 From _Vaclav Kotesovec_, Jun 20 2020: (Start)
%F A326306 Dirichlet g.f.: zeta(s) * Product_{primes p} (1 + 1/(p^s - p)).
%F A326306 Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) + p^(-s)). (End)
%F A326306 Multiplicative with a(p^e) = 1 + (p^e-1)/(p-1). - _Amiram Eldar_, Oct 14 2020
%F A326306 From _Richard L. Ollerton_, May 07 2021: (Start)
%F A326306 a(n) = Sum_{k=1..n} mu(n/gcd(n,k))*sigma(gcd(n,k)).
%F A326306 a(n) = Sum_{k=1..n} mu(gcd(n,k))*sigma(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
%t A326306 Table[Sum[d/Last[Select[Divisors[d], SquareFreeQ]], {d, Divisors[n]}], {n, 1, 85}]
%t A326306 Table[Sum[MoebiusMu[n/d] EulerPhi[n/d] DivisorSigma[1, d], {d, Divisors[n]}], {n, 1, 85}]
%t A326306 f[p_, e_] := 1 + (p^e-1)/(p-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Oct 14 2020 *)
%o A326306 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + X)/(1 - X)/(1 - p*X))[n], ", ")) \\ _Vaclav Kotesovec_, Jun 14 2020
%Y A326306 Cf. A000010, A000079 (fixed points), A000203, A003557, A007947, A008683, A098108 (parity of a(n)), A191750, A300717, A335032.
%K A326306 nonn,mult,easy
%O A326306 1,2
%A A326306 _Ilya Gutkovskiy_, Oct 17 2019