cp's OEIS Frontend

A007406 Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^2.

%I A007406 M4004 #119 Feb 16 2025 08:32:31
%S A007406 1,5,49,205,5269,5369,266681,1077749,9778141,1968329,239437889,
%T A007406 240505109,40799043101,40931552621,205234915681,822968714749,
%U A007406 238357395880861,238820721143261,86364397717734821,17299975731542641,353562301485889,354019312583809,187497409728228241
%N A007406 Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^2.
%C A007406 By Wolstenholme's theorem, p divides a(p-1) for prime p > 3. - _T. D. Noe_, Sep 05 2002
%C A007406 Also p divides a( (p-1)/2 ) for prime p > 3. - _Alexander Adamchuk_, Jun 07 2006
%C A007406 The rationals a(n)/A007407(n) converge to Zeta(2) = (Pi^2)/6 = 1.6449340668... (see the decimal expansion A013661).
%C A007406 For the rationals a(n)/A007407(n), n >= 1, see the W. Lang link under A103345 (case k=2).
%C A007406 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
%C A007406 Denominator of the harmonic mean of the first n squares. - _Colin Barker_, Nov 13 2014
%C A007406 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
%C A007406 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
%C A007406 From _Peter Bala_, Feb 16 2022: (Start)
%C A007406 Sum_{k = 1..n} 1/k^2 = 1 + (1 - 1/2^2)*(n-1)/(n+1) - (1/2^2 - 1/3^2)*(n-1)*(n-2)/((n+1)*(n+2)) + (1/3^2 - 1/4^2)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) - (1/4^2 - 1/5^2)*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) + .... Cf. A082687 and A120778.
%C A007406 This identity allows us to extend the definition of Sum_{k = 1..n} 1/k^2 to non-integral values of n. (End)
%C A007406 Numerators of the Eulerian numbers T(-2,k) for k = 0,1..., if T(n,k) is extended to negative n by the recurrence T(n,k) = (k+1)*T(n-1,k) + (n-k)*T(n-1,k-1) (indexed as in A173018). - _Michael J. Collins_, Oct 10 2024
%D A007406 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007406 Seiichi Manyama, <a href="/A007406/b007406.txt">Table of n, a(n) for n = 1..1152</a> (terms 1..200 from T. D. Noe)
%H A007406 Stephen Crowley, <a href="https://arxiv.org/abs/1207.1126">Two New Zeta Constants: Fractal String, Continued Fraction, and Hypergeometric Aspects of the Riemann Zeta Function</a>, arXiv:1207.1126 [math.NT], 2012.
%H A007406 Romeo Mestrovic, <a href="http://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv:1111.3057 [math.NT], 2011.
%H A007406 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha103.htm">Factorizations of many number sequences</a>
%H A007406 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha129.htm">Factorizations of many number sequences</a>
%H A007406 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha1291.htm">Factorizations of many number sequences</a>
%H A007406 D. Y. Savio, E. A. Lamagna and S.-M. Liu, <a href="http://dx.doi.org/10.1007/978-1-4613-9647-5_2">Summation of harmonic numbers</a>, pp. 12-20 of E. Kaltofen and S. M. Watt, editors, Computers and Mathematics, Springer-Verlag, NY, 1989.
%H A007406 M. D. Schmidt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Schmidt/multifact.html">Generalized j-Factorial Functions, Polynomials, and Applications </a>, J. Int. Seq. 13 (2010), 10.6.7, Section 4.3.2.
%H A007406 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WolstenholmesTheorem.html">Wolstenholme's Theorem</a>
%H A007406 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WolstenholmeNumber.html">Wolstenholme Number</a>
%F A007406 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
%F A007406 G.f. for rationals a(n)/A007407(n), n >= 1: polylog(2,x)/(1-x).
%F A007406 a(n) = Numerator of (Pi^2)/6 - Zeta(2,n). - _Artur Jasinski_, Mar 03 2010
%p A007406 a:= n-> numer(add(1/i^2, i=1..n)): seq(a(n), n=1..24);  # _Zerinvary Lajos_, Mar 28 2007
%t A007406 a[n_] := If[ n<1, 0, Numerator[HarmonicNumber[n, 2]]]; Table[a[n], {n, 100}]
%t A007406 Numerator[HarmonicNumber[Range[20],2]] (* _Harvey P. Dale_, Jul 06 2014 *)
%o A007406 (PARI) {a(n) = if( n<1, 0, numerator( sum( k=1, n, 1 / k^2 ) ) )} /* _Michael Somos_, Jan 16 2011 */
%o A007406 (Haskell)
%o A007406 import Data.Ratio ((%), numerator)
%o A007406 a007406 n = a007406_list !! (n-1)
%o A007406 a007406_list = map numerator $ scanl1 (+) $ map (1 %) $ tail a000290_list
%o A007406 -- _Reinhard Zumkeller_, Jul 06 2012
%o A007406 (Magma) [Numerator(&+[1/k^2:k in [1..n]]):n in [1..23]]; // _Marius A. Burtea_, Aug 02 2019
%Y A007406 Cf. A001008, A007407 (denominators), A000290, A082687, A120778.
%Y A007406 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
%K A007406 nonn,frac,easy,nice
%O A007406 1,2
%A A007406 _N. J. A. Sloane_, _Mira Bernstein_