A046970 Dirichlet inverse of the Jordan function J_2 (A007434).

1, -3, -8, -3, -24, 24, -48, -3, -8, 72, -120, 24, -168, 144, 192, -3, -288, 24, -360, 72, 384, 360, -528, 24, -24, 504, -8, 144, -840, -576, -960, -3, 960, 864, 1152, 24, -1368, 1080, 1344, 72, -1680, -1152, -1848, 360, 192, 1584, -2208, 24, -48, 72, 2304, 504, -2808, 24, 2880, 144, 2880, 2520, -3480, -576

Offset

1,2

Comments

B(n+2) = -B(n)*((n+2)*(n+1)/(4*Pi^2))*z(n+2)/z(n) = -B(n)*((n+2)*(n+1)/(4*Pi^2)) * Sum_{j>=1} a(j)/j^(n+2).

Apart from signs also Sum_{d|n} core(d)^2*mu(n/d) where core(x) is the squarefree part of x. - Benoit Cloitre, May 31 2002

References

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, pp. 805-811.

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, p. 48.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.

Paul W. Oxby, A Function Based on Chebyshev Polynomials as an Alternative to the Sinc Function in FIR Filter Design, arXiv:2011.10546 [eess.SP], 2020.

Wikipedia, Riemann zeta function.

Formula

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

a(n) = Sum_{d|n} mu(d)*d^2.

abs(a(n)) = Product_{p prime divides n} (p^2 - 1). - Jon Perry, Aug 24 2010

From Wolfdieter Lang, Jun 16 2011: (Start)

Dirichlet g.f.: zeta(s)/zeta(s-2).

a(n) = J_{-2}(n)*n^2, with the Jordan function J_k(n), with J_k(1):=1. See the Apostol reference, p. 48. exercise 17. (End)

a(prime(n)) = -A084920(n). - R. J. Mathar, Aug 28 2011

G.f.: Sum_{k>=1} mu(k)*k^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 15 2017

a(n) = Sum_{d divides n} d * (sigma_1(d))^(-1) * sigma_1(n/d), where (sigma_1(n))^(-1) = A046692(n) denotes the Dirichlet inverse of sigma_1(n). - Peter Bala, Jan 26 2024

a(n) = A076479(n) * A322360(n). - Amiram Eldar, Feb 02 2024

Example

a(3) = -8 because the divisors of 3 are {1, 3} and mu(1)*1^2 + mu(3)*3^2 = -8.
a(4) = -3 because the divisors of 4 are {1, 2, 4} and mu(1)*1^2 + mu(2)*2^2 + mu(4)*4^2 = -3.
E.g., a(15) = (3^2 - 1) * (5^2 - 1) = 8*24 = 192. - Jon Perry, Aug 24 2010
G.f. = x - 3*x^2 - 8*x^3 - 3*x^4 - 24*x^5 + 24*x^6 - 48*x^7 - 3*x^8 - 8*x^9 + ...

Maple

Jinvk := proc(n, k) local a, f, p ; a := 1 ; for f in ifactors(n)[2] do p := op(1, f) ; a := a*(1-p^k) ; end do: a ; end proc:
A046970 := proc(n) Jinvk(n, 2) ; end proc: # R. J. Mathar, Jul 04 2011

Mathematica

muDD[d_] := MoebiusMu[d]*d^2; Table[Plus @@ muDD[Divisors[n]], {n, 60}] (Lopez)
Flatten[Table[{ x = FactorInteger[n]; p = 1; For[i = 1, i <= Length[x], i++, p = p*(1 - x[[i]][[1]]^2)]; p}, {n, 1, 50, 1}]] (* Jon Perry, Aug 24 2010 *)
a[ n_] := If[ n < 1, 0, Sum[ d^2 MoebiusMu[ d], {d, Divisors @ n}]]; (* Michael Somos, Jan 11 2014 *)
a[ n_] := If[ n < 2, Boole[ n == 1], Times @@ (1 - #[[1]]^2 & /@ FactorInteger @ n)]; (* Michael Somos, Jan 11 2014 *)

Prog

(PARI) A046970(n)=sumdiv(n, d, d^2*moebius(d)) \\ Benoit Cloitre
(Haskell)
a046970 = product . map ((1 -) . (^ 2)) . a027748_row
-- Reinhard Zumkeller, Jan 19 2012
(PARI) {a(n) = if( n<1, 0, direuler( p=2, n, (1 - X*p^2) / (1 - X))[n])}; /* Michael Somos, Jan 11 2014 */
(Python)
from math import prod
from sympy import primefactors
def A046970(n): return prod(1-p**2 for p in primefactors(n)) # Chai Wah Wu, Sep 08 2023

Crossrefs

Dirichlet inverse of Jordan totient function J_r(n): A023900 (r = 1), A063453(r = 3), A189922 (r = 4).

Cf. A007434, A027641, A027642, A027748, A046692, A076479, A084920, A322360.

Sequence in context: A144457 A220138 A146975 * A322360 A058936 A280369

Adjacent sequences: A046967 A046968 A046969 * A046971 A046972 A046973

Keyword

sign,easy,mult

Author

Douglas Stoll, dougstoll(AT)email.msn.com

Extensions

Corrected and extended by Vladeta Jovovic, Jul 25 2001

Additional comments from Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Jul 01 2005

Status

approved