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A046970
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Dirichlet inverse of the Jordan function J_2 (A007434).
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8
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| B(n+2) = -B(n)*((n+2)*(n+1)/(4pi^2))*z(n+2)/z(n) = -B(n)*((n+2)*(n+1)/(4pi^2))*Sum(j=1, infinity) [ 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 (benoit7848c(AT)orange.fr), May 31 2002
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REFERENCES
| M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, pp. 805-811.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, p. 48.
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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.
Wikipedia, Riemann zeta function.
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FORMULA
| Multiplicative with a(p^e) = 1-p^2. a(n) = Sum_{d|n} mu(d)*d^2.
a(n) = product[p prime divides n, p^2-1] (gives unsigned version) [From Jon Perry (jonperrydc(AT)btinternet.com), 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
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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 [From Jon Perry (jonperrydc(AT)btinternet.com), Aug 24 2010]
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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
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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*(x[[i]][[1]]^2 - 1)]; p}, {n, 1, 50, 1}]] [From Jon Perry (jonperrydc(AT)btinternet.com), Aug 24 2010]
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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
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CROSSREFS
| Cf. A027641, A027642, A063453, A023900.
Cf. A027748.
Sequence in context: A035292 A144457 A146975 * A058936 A002017 A118582
Adjacent sequences: A046967 A046968 A046969 * A046971 A046972 A046973
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KEYWORD
| sign,easy,mult
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AUTHOR
| Douglas Stoll, dougstoll(AT)email.msn.com
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EXTENSIONS
| Corrected and extended by Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 25 2001
Additional comments from Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Jul 01 2005
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