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%I #67 Feb 03 2024 10:16:31
%S 1,-3,-8,-3,-24,24,-48,-3,-8,72,-120,24,-168,144,192,-3,-288,24,-360,
%T 72,384,360,-528,24,-24,504,-8,144,-840,-576,-960,-3,960,864,1152,24,
%U -1368,1080,1344,72,-1680,-1152,-1848,360,192,1584,-2208,24,-48,72,2304,504,-2808,24,2880,144,2880,2520,-3480,-576
%N Dirichlet inverse of the Jordan function J_2 (A007434).
%C 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).
%C 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
%D M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, pp. 805-811.
%D Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, p. 48.
%H Reinhard Zumkeller, <a href="/A046970/b046970.txt">Table of n, a(n) for n = 1..10000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H P. G. Brown, <a href="http://www.jstor.org/stable/3621931">Some comments on inverse arithmetic functions</a>, Math. Gaz. 89 (516) (2005) 403-408.
%H Paul W. Oxby, <a href="https://arxiv.org/abs/2011.10546">A Function Based on Chebyshev Polynomials as an Alternative to the Sinc Function in FIR Filter Design</a>, arXiv:2011.10546 [eess.SP], 2020.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Riemann_zeta_function">Riemann zeta function</a>.
%F Multiplicative with a(p^e) = 1 - p^2.
%F a(n) = Sum_{d|n} mu(d)*d^2.
%F abs(a(n)) = Product_{p prime divides n} (p^2 - 1). - _Jon Perry_, Aug 24 2010
%F From _Wolfdieter Lang_, Jun 16 2011: (Start)
%F Dirichlet g.f.: zeta(s)/zeta(s-2).
%F 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)
%F a(prime(n)) = -A084920(n). - _R. J. Mathar_, Aug 28 2011
%F G.f.: Sum_{k>=1} mu(k)*k^2*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Jan 15 2017
%F 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
%F a(n) = A076479(n) * A322360(n). - _Amiram Eldar_, Feb 02 2024
%e a(3) = -8 because the divisors of 3 are {1, 3} and mu(1)*1^2 + mu(3)*3^2 = -8.
%e 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 E.g., a(15) = (3^2 - 1) * (5^2 - 1) = 8*24 = 192. - _Jon Perry_, Aug 24 2010
%e 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 + ...
%p 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:
%p A046970 := proc(n) Jinvk(n,2) ; end proc: # _R. J. Mathar_, Jul 04 2011
%t muDD[d_] := MoebiusMu[d]*d^2; Table[Plus @@ muDD[Divisors[n]], {n, 60}] (Lopez)
%t 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 *)
%t a[ n_] := If[ n < 1, 0, Sum[ d^2 MoebiusMu[ d], {d, Divisors @ n}]]; (* _Michael Somos_, Jan 11 2014 *)
%t a[ n_] := If[ n < 2, Boole[ n == 1], Times @@ (1 - #[[1]]^2 & /@ FactorInteger @ n)]; (* _Michael Somos_, Jan 11 2014 *)
%o (PARI) A046970(n)=sumdiv(n,d,d^2*moebius(d)) \\ _Benoit Cloitre_
%o (Haskell)
%o a046970 = product . map ((1 -) . (^ 2)) . a027748_row
%o -- _Reinhard Zumkeller_, Jan 19 2012
%o (PARI) {a(n) = if( n<1, 0, direuler( p=2, n, (1 - X*p^2) / (1 - X))[n])}; /* _Michael Somos_, Jan 11 2014 */
%o (Python)
%o from math import prod
%o from sympy import primefactors
%o def A046970(n): return prod(1-p**2 for p in primefactors(n)) # _Chai Wah Wu_, Sep 08 2023
%Y Dirichlet inverse of Jordan totient function J_r(n): A023900 (r = 1), A063453(r = 3), A189922 (r = 4).
%Y Cf. A007434, A027641, A027642, A027748, A046692, A076479, A084920, A322360.
%K sign,easy,mult
%O 1,2
%A Douglas Stoll, dougstoll(AT)email.msn.com
%E Corrected and extended by _Vladeta Jovovic_, Jul 25 2001
%E Additional comments from Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Jul 01 2005
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