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 = x*y with 1 <= x <= n, 1 <= y <= n, x and y squarefree). - Benoit Cloitre, Jan 01 2003
REFERENCES
Tom 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
Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Yu Hin (Gary) Au, Decompositions of Unit Hypercubes and the Reversion of a Generalized Möbius Series, arXiv:2205.03680 [math.CO], 2022.
Enrique Pérez Herrero, Mathematica Package for Piltz Divisor Functions.
Enrique Pérez Herrero, Mathematica Package for Piltz Divisor Functions.
Adolf Piltz, Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, Doctoral Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1881; the k-th Piltz function tau_k(n) is denoted by phi(n,k) and its recurrence and Dirichlet series appear on p. 6.
N. J. A. Sloane, Transforms.
Wikipedia, Adolf Piltz.
FORMULA
Dirichlet g.f.: 1/zeta(s)^2.
Multiplicative function with a(p^e) = binomial(2, e)*(-1)^e for p prime and e >= 0.
a(n) = Sum_{d|n} mu(d)*mu(n/d). - Benoit Cloitre, Apr 05 2002
a(n^2) = A008683(n)^2. a(A005117(n)) = (-2)^A001221(A005117(n)). - Enrique Pérez Herrero, Jun 27 2011 [Misrendering of contribution rectified by Peter Munn, Mar 06 2020]
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. The conjecture is correct because 1+Sum_{e>=1} binomial(2,e)/p^(e*s) = (p^s+1)^2/p^2s, whose product over p is zeta(s)^2/zeta(2s)^2. - Michael Shamos
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} tau(k)*A(x^k), where tau = A000005. - Ilya Gutkovskiy, May 11 2019
Sum_{k=1..n} abs(a(k)) ~ (n/zeta(2)^2) * (log(n) + 2*gamma - 1 - 4*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 24 2023
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 + ...
We have a(3^1) = C(2, 1)*(-1)^1 = -2, a(3^2) = C(2, 2)*(-1)^2 = 1, and a(3^m) = C(2, m)*(-1)^m = 0 for m >= 3. - Petros Hadjicostas, Jun 07 2019
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
(Python)
from math import prod, comb
from sympy import factorint
def A007427(n): return prod(-comb(2, e) if e&1 else comb(2, e) for e in factorint(n).values()) # Chai Wah Wu, Jul 05 2024
CROSSREFS
KEYWORD
sign,easy,nice,mult
AUTHOR
EXTENSIONS
Added a proof of Stephan's conjecture about the Dirichlet g.f. of |a(n)|.
STATUS
approved