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A007407
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a(n) = denominator of Sum_{k=1..n} 1/k^2.
(Formerly M3661)
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49
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1, 4, 36, 144, 3600, 3600, 176400, 705600, 6350400, 1270080, 153679680, 153679680, 25971865920, 25971865920, 129859329600, 519437318400, 150117385017600, 150117385017600, 54192375991353600, 10838475198270720, 221193371393280
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Seiichi Manyama, Table of n, a(n) for n = 1..1152 (terms 1..200 from T. D. Noe)
D. Y. Savio, E. A. Lamagna and S.-M. Liu, Summation of harmonic numbers, pp. 12-20 of E. Kaltofen and S. M. Watt, editors, Computers and Mathematics, Springer-Verlag, NY, 1989.
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FORMULA
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a(n) = denominator of (Pi^2)/6 - zeta(2, x). - Artur Jasinski, Mar 03 2010
a(n) = A001044(n) / gcd(A001819(n), A001044(n)). - Daniel Suteu, Dec 25 2016
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EXAMPLE
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1/1^2 + 1/2^2 + 1/3^2 = 1/1 + 1/4 + 1/9 = 49/36, so a(3) = 36. - Jon E. Schoenfield, Dec 26 2014
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MAPLE
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ZL:=n->sum(1/i^2, i=2..n): a:=n->floor(denom(ZL(n))): seq(a(n), n=1..21); # Zerinvary Lajos, Mar 28 2007
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MATHEMATICA
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s=0; lst={}; Do[s+=n^2/n^4; AppendTo[lst, Denominator[s]], {n, 3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[Denominator[Pi^2/6 - Zeta[2, x]], {x, 1, 22}] (* Artur Jasinski, Mar 03 2010 *)
Denominator[Accumulate[1/Range[30]^2]] (* Harvey P. Dale, Nov 08 2012 *)
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PROG
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(Haskell)
import Data.Ratio ((%), denominator)
a007407 n = a007407_list !! (n-1)
a007407_list = map denominator $
scanl1 (+) $ map (1 %) $ tail a000290_list
-- Reinhard Zumkeller, Jul 06 2012
(PARI) a(n)=denominator(sum(k=1, n, 1/k^2)) \\ Charles R Greathouse IV, Nov 20 2012
(Python)
from fractions import Fraction
def A007407(n): return sum(Fraction(1, k**2) for k in range(1, n+1)).denominator # Chai Wah Wu, Apr 03 2021
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CROSSREFS
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Cf. A007406 (numerators), A000290, A035166.
Sequence in context: A334580 A068589 A120077 * A051418 A069046 A065886
Adjacent sequences: A007404 A007405 A007406 * A007408 A007409 A007410
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KEYWORD
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nonn,easy,frac,nice
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AUTHOR
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N. J. A. Sloane, Mira Bernstein
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STATUS
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approved
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