OFFSET
1,2
COMMENTS
Dirichlet inverse of A007425. - R. J. Mathar, Jul 15 2010
abs(a(n)) is the number of ways to write n=xyz where x,y,z are squarefree numbers. - Benoit Cloitre, Jan 02 2018
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms
FORMULA
Multiplicative with a(p^e) = (3 choose e) (-1)^e.
Dirichlet g.f.: 1/zeta(s)^3.
From Enrique Pérez Herrero, Jul 12 2010: (Start)
a(n^3) = A008683(n).
a(A046101(n)) = 0. - Enrique Pérez Herrero, Sep 07 2017
a(n) = Sum_{a*b*c=n} mu(a)*mu(b)*mu(c). - Benedict W. J. Irwin, Mar 02 2022
MAPLE
möbius := proc(a) local b, i, mo: b := NULL:
mo := (m, n) -> `if`(irem(m, n) = 0, numtheory:-mobius(m/n), 0);
for i to nops(a) do b := b, add(mo(i, j)*a[j], j=1..i) od: [b] end:
(möbius@@3)([1, seq(0, i=1..77)]); # Peter Luschny, Sep 08 2017
MATHEMATICA
tau[1, n_Integer]:=1; SetAttributes[tau, Listable];
tau[k_Integer, n_Integer]:=Plus@@(tau[k-1, Divisors[n]])/; k > 1;
tau[k_Integer, n_Integer]:=Plus@@(tau[k+1, Divisors[n]]*MoebiusMu[n/Divisors[n]]); k<1;
A007428[n_]:=tau[ -3, n]; (* Enrique Pérez Herrero, Jul 12 2010 *)
a[n_] := Which[n==1, 1, PrimeQ[n], -3, True, Times @@ Map[Function[e, Binomial[3, e] (-1)^e], FactorInteger[n][[All, 2]]]];
Array[a, 100] (* Jean-François Alcover, Jun 20 2018 *)
PROG
(Haskell)
a007428 n = product
[a007318' 3 e * cycle [1, -1] !! fromIntegral e | e <- a124010_row n]
-- Reinhard Zumkeller, Oct 09 2013
(PARI) a(n) = {my(f=factor(n)); for (k=1, #f~, e = f[k, 2]; f[k, 1] = binomial(3, e)*(-1)^e; f[k, 2] = 1); factorback(f); } \\ Michel Marcus, Jan 03 2018
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - X)^3)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021
CROSSREFS
KEYWORD
sign,easy,nice,mult
AUTHOR
STATUS
approved