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A007406 Wolstenholme numbers: numerator of Sum 1/k^2, k = 1..n.
(Formerly M4004)
64
1, 5, 49, 205, 5269, 5369, 266681, 1077749, 9778141, 1968329, 239437889, 240505109, 40799043101, 40931552621, 205234915681, 822968714749, 238357395880861, 238820721143261, 86364397717734821, 17299975731542641 (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).

Numbers n such that a(n) is prime are listed in A111354[n] = {2,7,13,19,121,188,252,368,605,745,1085,1127,1406,...}. Primes in a(n) are listed in A123751[n] = {5,266681,40799043101,86364397717734821,...}. - Alexander Adamchuk, Oct 11 2006

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

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..200

S. Crowley, Some Fractal String and Hypergeometric Aspects of the Riemann Zeta Function, 2012. - N. J. A. Sloane, Jun 14 2012

R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057, 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.

Eric Weisstein's World of Mathematics, Wolstenholme's Theorem

Eric Weisstein's World of Mathematics, Wolstenholme Number

FORMULA

Sum[1/k^2, {k, 1, n}] = Sqrt[Sum[Sum[1/(i*j)^2, {i, 1, n}], {j, 1, n}]]. - 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

ZL:=n->sum(1/i^2, i=1..n): a:=n->floor(numer(ZL(n))): seq(a(n), n=1..20); # 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

CROSSREFS

Cf. A001008, A007407, A111354, A123751, A000290.

Sequence in context: A063429 A266127 A183333 * A196326 A273385 A058927

Adjacent sequences:  A007403 A007404 A007405 * A007407 A007408 A007409

KEYWORD

nonn,frac,easy,nice

AUTHOR

N. J. A. Sloane, Mira Bernstein

STATUS

approved

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Last modified March 29 18:39 EDT 2017. Contains 284273 sequences.