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A007427 Moebius transform applied twice to sequence 1,0,0,0,....
(Formerly M0198)
21
1, -2, -2, 1, -2, 4, -2, 0, 1, 4, -2, -2, -2, 4, 4, 0, -2, -2, -2, -2, 4, 4, -2, 0, 1, 4, 0, -2, -2, -8, -2, 0, 4, 4, 4, 1, -2, 4, 4, 0, -2, -8, -2, -2, -2, 4, -2, 0, 1, -2, 4, -2, -2, 0, 4, 0, 4, 4, -2, 4, -2, 4, -2, 0, 4, -8, -2, -2, 4, -8, -2, 0, -2, 4, -2, -2, 4, -8, -2, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

|a(n)| is the number of ways to write n as a product of 2 squarefree numbers (i.e. number of ways to write n = xy 1<=x<=n 1<=y<=n, x and y squarefree). - Benoit Cloitre, Jan 01 2003

REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 30.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=1..1000

E. Pérez Herrero, Mathematica Package for Piltz Divisor Functions

E. Pérez Herrero, Mathematica Package for Piltz Divisor Functions

N. J. A. Sloane, Transforms

FORMULA

Dirichlet g.f.: 1/zeta(s)^2.

Multiplicative with a(p^e) = (2 choose e) (-1)^e.

a(n) = sumd( d divides n, mu(d)*mu(n/d)). - Benoit Cloitre, Apr 05 2002

a(n^2) = A008683(n)^2. a(n) = (-2)^A001221(A005117(n)). - Enrique Pérez Herrero, Jun 27 2011

a(n) is the Dirichlet inverse of A000005, which means a(n) = -sum_{d|n, d<n} A000005(n/d)*a(d). Enrique Pérez Herrero, Jan 19 2013

a(n) = 0 if n is not cubefree: A046099, otherwise sign(a(n))= lambda(n), where lambda is A008836. - Enrique Pérez Herrero, Jan 19 2013

Dirichlet g.f. of |a(n)|: zeta(s)^2/zeta(2s)^2 (conjectured). - Ralf Stephan, Jul 05 2013

a(n) = sum(A225817(n,k)*A225817(n,n+1-k): k=1..A000005(n)). - Reinhard Zumkeller, Jul 30 2013

EXAMPLE

G.f. = x - 2*x^2 - 2*x^3 + x^4 - 2*x^5 + 4*x^6 - 2*x^7 + x^9 + 4*x^10 + ...

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@@2)([1, seq(0, i=1..80)]); # Peter Luschny, Sep 08 2017

MATHEMATICA

f[n_] := Plus @@ Times @@@ (MoebiusMu[{#, n/#}] & /@ Divisors@n); Array[f, 105] (* Robert G. Wilson v *)

a[n_] := DivisorSum[n, MoebiusMu[#]*MoebiusMu[n/#]&]; Array[a, 80] (* Jean-François Alcover, Dec 01 2015 *)

PROG

(PARI) {a(n) = if( n<1, 0, direuler(p=2, n, (1 - X)^2)[n])}; /* Michael Somos, Nov 15 2002 */

(PARI) {a(n) = if(n<1, 0, sumdiv(n, d, moebius(d) * moebius(n/d)))}; /* Michael Somos, Nov 15 2002 */

(PARI) a(n)=if(n>1, my(f=factor(n)[, 2], s=sum(i=1, #f, f[i]==1)); if(vecmax(f)>2, 0, (-1)^s<<s), 1) \\ Charles R Greathouse IV, Jun 28 2011

(Haskell)

a007427 n = sum $ zipWith (*) mds $ reverse mds where

   mds = a225817_row n

-- Reinhard Zumkeller, Jul 30 2013

CROSSREFS

Dirichlet inverse of A000005, Mobius transform of A008683.

Cf. A063524, A007428, A005117.

Sequence in context: A127677 A238009 A231145 * A048106 A228441 A156260

Adjacent sequences:  A007424 A007425 A007426 * A007428 A007429 A007430

KEYWORD

sign,easy,nice,mult

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified October 23 23:22 EDT 2017. Contains 293833 sequences.