A007428 Moebius transform applied thrice to sequence 1,0,0,0,....

1, -3, -3, 3, -3, 9, -3, -1, 3, 9, -3, -9, -3, 9, 9, 0, -3, -9, -3, -9, 9, 9, -3, 3, 3, 9, -1, -9, -3, -27, -3, 0, 9, 9, 9, 9, -3, 9, 9, 3, -3, -27, -3, -9, -9, 9, -3, 0, 3, -9, 9, -9, -3, 3, 9, 3, 9, 9, -3, 27, -3, 9, -9, 0, 9, -27, -3, -9, 9, -27, -3, -3, -3, 9, -9, -9, 9, -27

Offset: 1

Author

N. J. A. Sloane

Keywords

• sign
• easy
• nice
• mult

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

Crossrefs

Consecutive nested Dirichlet convolution: A063524, A008683 or A007427. - Enrique Pérez Herrero, Jul 12 2010

Cf. A124010.

Programs

Haskell

a007428 n = product
   [a007318' 3 e * cycle [1,-1] !! fromIntegral e | e <- a124010_row n]
-- Reinhard Zumkeller, Oct 09 2013

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 *)

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

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(s) = (-3)^A001221(s) provided s is a squarefree number (A005117). (End)

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