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 A007406 Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^2. (Formerly M4004) 72
 1, 5, 49, 205, 5269, 5369, 266681, 1077749, 9778141, 1968329, 239437889, 240505109, 40799043101, 40931552621, 205234915681, 822968714749, 238357395880861, 238820721143261, 86364397717734821, 17299975731542641, 353562301485889, 354019312583809, 187497409728228241 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS By Wolstenholme's theorem, p divides a(p-1) for prime p > 3. - T. D. Noe, Sep 05 2002 Also p divides a( (p-1)/2 ) for prime p > 3. - Alexander Adamchuk, Jun 07 2006 The rationals a(n)/A007407(n) converge to Zeta(2) = (Pi^2)/6 = 1.6449340668... (see the decimal expansion A013661). For the rationals a(n)/A007407(n), n >= 1, see the W. Lang link under A103345 (case k=2). See the Wolfdieter Lang link under A103345 on Zeta(k, n) with the rationals for k=1..10, g.f.s and polygamma formulas. - Wolfdieter Lang, Dec 03 2013 Denominator of the harmonic mean of the first n squares. - Colin Barker, Nov 13 2014 Conjecture: for n > 3, gcd(n, a(n-1)) = A089026(n). Checked up to n = 10^5. - Amiram Eldar and Thomas Ordowski, Jul 28 2019 True if n is prime, by Wolstenholme's theorem. It remains to show that gcd(n, a(n-1)) = 1 if n > 3 is composite. - Jonathan Sondow, Jul 29 2019 REFERENCES 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..1152 (terms 1..200 from T. D. Noe) Stephen Crowley, Two New Zeta Constants: Fractal String, Continued Fraction, and Hypergeometric Aspects of the Riemann Zeta Function, arXiv:1207.1126 [math.NT], 2012. R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011. Hisanori Mishima, Factorizations of many number sequences Hisanori Mishima, Factorizations of many number sequences Hisanori Mishima, Factorizations of many number sequences 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. M. D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Int. Seq. 13 (2010), 10.6.7, Section 4.3.2. Eric Weisstein's World of Mathematics, Wolstenholme's Theorem Eric Weisstein's World of Mathematics, Wolstenholme Number FORMULA Sum_{k=1..n} 1/k^2 = sqrt(Sum_{j=1..n} Sum_{i=1..n} 1/(i*j)^2). - Alexander Adamchuk, Oct 26 2004 G.f. for rationals a(n)/A007407(n), n >= 1: polylog(2,x)/(1-x). a(n) = Numerator of (Pi^2)/6 - Zeta(2,n). - Artur Jasinski, Mar 03 2010 MAPLE a:= n-> numer(add(1/i^2, i=1..n)): seq(a(n), n=1..24);  # Zerinvary Lajos, Mar 28 2007 MATHEMATICA a[n_] := If[ n<1, 0, Numerator[HarmonicNumber[n, 2]]]; Table[a[n], {n, 100}] Numerator[HarmonicNumber[Range[20], 2]] (* Harvey P. Dale, Jul 06 2014 *) PROG (PARI) {a(n) = if( n<1, 0, numerator( sum( k=1, n, 1 / k^2 ) ) )} /* Michael Somos, Jan 16 2011 */ (Haskell) import Data.Ratio ((%), numerator) a007406 n = a007406_list !! (n-1) a007406_list = map numerator \$ scanl1 (+) \$ map (1 %) \$ tail a000290_list -- Reinhard Zumkeller, Jul 06 2012 (MAGMA) [Numerator(&+[1/k^2:k in [1..n]]):n in [1..23]]; // Marius A. Burtea, Aug 02 2019 CROSSREFS Cf. A001008, A007407 (denominators), A000290. Numbers n such that a(n) is prime are listed in A111354. Primes in {a(n)} are listed in A123751. - Alexander Adamchuk, Oct 11 2006 Sequence in context: A183333 A299312 A300113 * A196326 A273385 A058927 Adjacent sequences:  A007403 A007404 A007405 * A007407 A007408 A007409 KEYWORD nonn,frac,easy,nice AUTHOR STATUS approved

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Last modified April 15 19:21 EDT 2021. Contains 342977 sequences. (Running on oeis4.)