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A007428 Moebius transform applied thrice to sequence 1,0,0,0,....
(Formerly M2271)
11
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 (list; graph; refs; listen; history; text; internal format)
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(s) = (-3)^A001221(s) provided s is a squarefree number (A005117). (End)

a(A046101(n)) = 0. - Enrique Pérez Herrero, Sep 07 2017

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

CROSSREFS

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

Cf. A124010.

Sequence in context: A078229 A222292 A245441 * A184099 A074816 A203564

Adjacent sequences:  A007425 A007426 A007427 * A007429 A007430 A007431

KEYWORD

sign,easy,nice,mult,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified June 25 13:41 EDT 2018. Contains 311908 sequences. (Running on oeis4.)