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A130054 Inverse Moebius transform of A023900.

%I A130054 #35 Feb 20 2022 01:40:45
%S A130054 1,0,-1,-1,-3,0,-5,-2,-3,0,-9,1,-11,0,3,-3,-15,0,-17,3,5,0,-21,2,-7,0,
%T A130054 -5,5,-27,0,-29,-4,9,0,15,3,-35,0,11,6,-39,0,-41,9,9,0,-45,3,-11,0,15,
%U A130054 11,-51,0,27,10,17,0,-57,-3,-59,0,15,-5,33,0,-65,15,21
%N A130054 Inverse Moebius transform of A023900.
%C A130054 Multiplicative because A023900 is. - _Andrew Howroyd_, Aug 03 2018
%H A130054 Amiram Eldar, <a href="/A130054/b130054.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Andrew Howroyd)
%F A130054 A126988 * A130054 = d(n), A000005: (1, 2, 2, 3, 2, 4, 2, 4, 3, 4, ...).
%F A130054 a(n) = Sum_{d|n} A023900(n/d). - _Andrew Howroyd_, Aug 03 2018
%F A130054 a(n) = Sum_{d|n} d*mu(d)*tau(n/d). - _Ridouane Oudra_, Nov 17 2019
%F A130054 From _Werner Schulte_, Sep 06 2020: (Start)
%F A130054 Multiplicative with a(p^e) = 1 - (p-1) * e for prime p and e >= 0.
%F A130054 Dirichlet g.f.: (zeta(s))^2 / zeta(s-1).
%F A130054 Dirichlet convolution with A062570 equals A001511.
%F A130054 Dirichlet convolution with A018804 equals A000203.
%F A130054 Dirichlet inverse of A007431. (End)
%F A130054 a(n) = 1 - Sum_{k=1..n-1} a(gcd(n,k)). - _Ilya Gutkovskiy_, Nov 06 2020
%p A130054 with(numtheory): seq(add(d*mobius(d)*tau(n/d), d in divisors(n)), n=1..60); # _Ridouane Oudra_, Nov 17 2019
%t A130054 b[n_] := Sum[d MoebiusMu[d], {d, Divisors[n]}];
%t A130054 a[n_] := Sum[b[n/d], {d, Divisors[n]}];
%t A130054 a /@ Range[1, 100] (* _Jean-François Alcover_, Sep 20 2019, from PARI *)
%t A130054 f[p_, e_] := 1-(p-1)*e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 23 2020 *)
%o A130054 (PARI) \\ here b(n) is A023900
%o A130054 b(n)={sumdivmult(n, d, d*moebius(d))}
%o A130054 a(n)={sumdiv(n, d, b(n/d))} \\ _Andrew Howroyd_, Aug 03 2018
%o A130054 (Magma) [&+[d*MoebiusMu(d)*NumberOfDivisors(n div d):d in Divisors(n)]:n in [1..70]]; // _Marius A. Burtea_, Nov 17 2019
%Y A130054 Row sums of A129691.
%Y A130054 Cf. A126988, A000005, A023900, A062570, A001511, A018804, A000203, A007431.
%K A130054 sign,mult,easy
%O A130054 1,5
%A A130054 _Gary W. Adamson_, May 04 2007
%E A130054 Name changed and terms a(11) and beyond from _Andrew Howroyd_, Aug 03 2018