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A130054
Inverse Moebius transform of A023900.
5
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, -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, 11, -51, 0, 27, 10, 17, 0, -57, -3, -59, 0, 15, -5, 33, 0, -65, 15, 21
OFFSET
1,5
COMMENTS
Multiplicative because A023900 is. - Andrew Howroyd, Aug 03 2018
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)
FORMULA
A126988 * A130054 = d(n), A000005: (1, 2, 2, 3, 2, 4, 2, 4, 3, 4, ...).
a(n) = Sum_{d|n} A023900(n/d). - Andrew Howroyd, Aug 03 2018
a(n) = Sum_{d|n} d*mu(d)*tau(n/d). - Ridouane Oudra, Nov 17 2019
From Werner Schulte, Sep 06 2020: (Start)
Multiplicative with a(p^e) = 1 - (p-1) * e for prime p and e >= 0.
Dirichlet g.f.: (zeta(s))^2 / zeta(s-1).
Dirichlet convolution with A062570 equals A001511.
Dirichlet convolution with A018804 equals A000203.
Dirichlet inverse of A007431. (End)
a(n) = 1 - Sum_{k=1..n-1} a(gcd(n,k)). - Ilya Gutkovskiy, Nov 06 2020
MAPLE
with(numtheory): seq(add(d*mobius(d)*tau(n/d), d in divisors(n)), n=1..60); # Ridouane Oudra, Nov 17 2019
MATHEMATICA
b[n_] := Sum[d MoebiusMu[d], {d, Divisors[n]}];
a[n_] := Sum[b[n/d], {d, Divisors[n]}];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019, from PARI *)
f[p_, e_] := 1-(p-1)*e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 23 2020 *)
PROG
(PARI) \\ here b(n) is A023900
b(n)={sumdivmult(n, d, d*moebius(d))}
a(n)={sumdiv(n, d, b(n/d))} \\ Andrew Howroyd, Aug 03 2018
(Magma) [&+[d*MoebiusMu(d)*NumberOfDivisors(n div d):d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Nov 17 2019
CROSSREFS
KEYWORD
sign,mult,easy
AUTHOR
Gary W. Adamson, May 04 2007
EXTENSIONS
Name changed and terms a(11) and beyond from Andrew Howroyd, Aug 03 2018
STATUS
approved