login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

N. J. A. Sloane, Mira Bernstein

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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