Search: id:A007406
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%I A007406 M4004
%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
%N A007406 Wolstenholme numbers: numerator of Sum 1/k^2, k = 1..n.
%C A007406 By Wolstenholme's theorem, p divides a(p-1) for prime p > 3. - T. D. 
               Noe (noe(AT)sspectra.com), Sep 05 2002
%C A007406 Also p divides a( (p-1)/2 ) for prime p > 3. - Alexander Adamchuk (alex(AT)kolmogorov.com), 
               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 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 (alex(AT)kolmogorov.com), Oct 11 2006
%D A007406 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007406 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.
%H A007406 T. D. Noe, <a href="b007406.txt">Table of n, a(n) for n=1..200</a>
%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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               WolstenholmesTheorem.html">Wolstenholme's Theorem</a>
%H A007406 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               WolstenholmeNumber.html">Wolstenholme Number</a>
%F A007406 Sum[1/k^2, {k, 1, n}] = Sqrt[Sum[Sum[1/(i*j)^2, {i, 1, n}], {j, 1, n}]] 
               - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 26 2004
%F A007406 G.f. for rationals a(n)/A007407(n), n>=1: polylog(2,x)/(1-x).
%p A007406 ZL:=n->sum(1/i^2, i=1..n): a:=n->floor(numer(ZL(n))): seq(a(n), n=1..20); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 28 2007
%t A007406 s=0;lst={};Do[s+=n^2/n^4;AppendTo[lst,Numerator[s]],{n,3*4!}];lst [From 
               Vladimir Orlovsky (4vladimir(AT)gmail.com), Jan 24 2009]
%Y A007406 Cf. A001008, A007407.
%Y A007406 Cf. A111354, A123751.
%Y A007406 Sequence in context: A108207 A127091 A063429 this_sequence A058927 A083224 
               A093188
%Y A007406 Adjacent sequences: A007403 A007404 A007405 this_sequence A007407 A007408 
               A007409
%K A007406 nonn,frac,easy,nice
%O A007406 1,2
%A A007406 N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein

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