A007427 Moebius transform applied twice to sequence 1,0,0,0,....
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
Offset: 1
Examples
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
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.
Crossrefs
Programs
-
Haskell
a007427 n = sum $ zipWith (*) mds $ reverse mds where mds = a225817_row n -- Reinhard Zumkeller, Jul 30 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@@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 *) -
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<
-
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
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, dA000005(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
Extensions
Added a proof of Stephan's conjecture about the Dirichlet g.f. of |a(n)|.
Comments