login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A007428 Moebius transform applied thrice to sequence 1,0,0,0,....
(Formerly M2271)
21
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

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

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

Cf. A124010.

Sequence in context: A222292 A245441 A333793 * A184099 A074816 A203564

Adjacent sequences: A007425 A007426 A007427 * A007429 A007430 A007431

KEYWORD

sign,easy,nice,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 5 09:12 EST 2022. Contains 358585 sequences. (Running on oeis4.)