A130054 - OEIS

%I #35 Feb 20 2022 01:40:45

%S 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 -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 11,-51,0,27,10,17,0,-57,-3,-59,0,15,-5,33,0,-65,15,21

%N Inverse Moebius transform of A023900.

%C Multiplicative because A023900 is. - _Andrew Howroyd_, Aug 03 2018

%H Amiram Eldar, <a href="/A130054/b130054.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Andrew Howroyd)

%F A126988 * A130054 = d(n), A000005: (1, 2, 2, 3, 2, 4, 2, 4, 3, 4, ...).

%F a(n) = Sum_{d|n} A023900(n/d). - _Andrew Howroyd_, Aug 03 2018

%F a(n) = Sum_{d|n} d*mu(d)*tau(n/d). - _Ridouane Oudra_, Nov 17 2019

%F From _Werner Schulte_, Sep 06 2020: (Start)

%F Multiplicative with a(p^e) = 1 - (p-1) * e for prime p and e >= 0.

%F Dirichlet g.f.: (zeta(s))^2 / zeta(s-1).

%F Dirichlet convolution with A062570 equals A001511.

%F Dirichlet convolution with A018804 equals A000203.

%F Dirichlet inverse of A007431. (End)

%F a(n) = 1 - Sum_{k=1..n-1} a(gcd(n,k)). - _Ilya Gutkovskiy_, Nov 06 2020

%p with(numtheory): seq(add(d*mobius(d)*tau(n/d), d in divisors(n)), n=1..60); # _Ridouane Oudra_, Nov 17 2019

%t b[n_] := Sum[d MoebiusMu[d], {d, Divisors[n]}];

%t a[n_] := Sum[b[n/d], {d, Divisors[n]}];

%t a /@ Range[1, 100] (* _Jean-François Alcover_, Sep 20 2019, from PARI *)

%t 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 (PARI) \\ here b(n) is A023900

%o b(n)={sumdivmult(n, d, d*moebius(d))}

%o a(n)={sumdiv(n, d, b(n/d))} \\ _Andrew Howroyd_, Aug 03 2018

%o (Magma) [&+[d*MoebiusMu(d)*NumberOfDivisors(n div d):d in Divisors(n)]:n in [1..70]]; // _Marius A. Burtea_, Nov 17 2019

%Y Row sums of A129691.

%Y Cf. A126988, A000005, A023900, A062570, A001511, A018804, A000203, A007431.

%K sign,mult,easy

%O 1,5

%A _Gary W. Adamson_, May 04 2007

%E Name changed and terms a(11) and beyond from _Andrew Howroyd_, Aug 03 2018