Search: id:A001008
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%I A001008 M2885 N1157
%S A001008 1,3,11,25,137,49,363,761,7129,7381,83711,86021,1145993,
%T A001008 1171733,1195757,2436559,42142223,14274301,275295799,55835135,
%U A001008 18858053,19093197,444316699,1347822955,34052522467,34395742267
%N A001008 Wolstenholme numbers: numerator of harmonic number H(n)=Sum_{i=1..n} 
               1/i.
%C A001008 H(n) is the maximal distance that a stack of n cards can project beyond 
               the edge of a table without toppling.
%C A001008 By Wolstenholme's theorem, p^2 divides a(p-1) for prime p > 3.
%C A001008 Comments from Alexander Adamchuk (alex(AT)kolmogorov.com), Dec 11 2006 
               (Start)
%C A001008 p divides a(p^2-1) for prime p>3.
%C A001008 p divides a((p-1)/2) for prime p = {1093, 3511, ...} = A001220(n) = Wieferich 
               primes p: p^2 divides 2^(p-1) - 1.
%C A001008 p divides a((p+1)/2) or a((p-3)/2) for prime p = {3, 29, 37, 3373, ...} 
               = A125854(n) that apart from the first term appears to coincide with 
               A121999(n) = {29, 37, 3373, ...} Primes p such that p^2 divides Sierpinski 
               number A014566[(p-1)/2].
%C A001008 a(n) is prime for n = {2, 3, 5, 8, 9, 21, 26, 41, 56, 62, 69, 79, 89, 
               91, ...} = A056903(n).
%C A001008 Corresponding primes a(n) are a(A056903(n)) = A067657(n) = {3, 11, 137, 
               761, 7129, 18858053, 34395742267, 85691034670497533, ...}. (End)
%C A001008 a(n+1)= numerator of amazing polynomial A[1,n](1) where amazing polynomial 
               A[genus 1,level n](m) is defined as Sum[m^(n - d)/d] d=1..n-1 Mathematica 
               procedure generating A[1,n](m)is: m =.; aa = {}; Do[k = 0; Do[k = 
               k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, k], {r, 1, 20}]; aa 
               [From Artur Jasinski (grafix(AT)csl.pl), Oct 16 2008]
%D A001008 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001008 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001008 Hongwei Chen, Evaluations of Some Variant Euler Sums, Journal of Integer 
               Sequences, Vol. 9 (2006), Article 06.2.3.
%D A001008 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 1990, p. 259.
%D A001008 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 
               3rd ed., Oxford Univ. Press, 1954, page 347.
%D A001008 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, 
               MA, Vol. 1, p. 615.
%H A001008 T. D. Noe, <a href="b001008.txt">Table of n, a(n) for n = 1..200</a>
%H A001008 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/harmonic.html">
               Harmonic numbers and the book-stacking problem</a>
%H A001008 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha126.htm">Factorizations of many number sequences</a>
%H A001008 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha127.htm">Factorizations of many number sequences</a>
%H A001008 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha128.htm">Factorizations of many number sequences</a>
%H A001008 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha1281.htm">Factorizations of many number sequences</a>
%H A001008 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha103.htm">Factorizations of many number sequences</a>
%H A001008 N. J. A. Sloane, <a href="a1008.gif">Illustration of initial terms</a>
%H A001008 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BookStackingProblem.html">Link to a section of The World of Mathematics.</
               a>
%H A001008 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HarmonicNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A001008 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               WolstenholmesTheorem.html">Wolstenholme's Theorem</a>
%H A001008 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HarmonicMean.html">Harmonic Mean</a>
%F A001008 H(n) ~ log n + gamma + O(1/n) [see for example Hardy and Wright, Th. 
               422.]
%F A001008 log n + gamma - 1/n < H(n) < log n + gamma + 1/n [follows easily from 
               Hardy and Wright, Th. 422] (David Applegate and N. J. A. Sloane (njas(AT)research.att.com), 
               Oct 14 2008)
%F A001008 G.f. for H(n) : log(1-x)/(x-1) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Jun 15 2003
%F A001008 H(n) = Sqrt[Sum[Sum[1/(i*j), {i, 1, n}], {j, 1, n}]] - Alexander Adamchuk 
               (alex(AT)kolmogorov.com), Oct 24 2004
%F A001008 a(n)=Numerator[EulerGamma/n + PolyGamma[0, 1 + n]/n] [From Artur Jasinski 
               (grafix(AT)csl.pl), Nov 02 2008]
%e A001008 H(n) = [ 1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520,
               ... ].
%p A001008 ZL:=n->sum(1/i, i=1..n): a:=n->floor(numer(ZL(n))): seq(a(n), n=1..26); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 28 2007
%t A001008 a = 1; b = 1; maxN = 26; s = 0; Numerator[ Table[ s += 1/(a*n + b), {n, 
               0, maxN} ]]
%t A001008 H(n) = Table[Sqrt[Sum[Sum[1/(i*j), {i, 1, n}], {j, 1, n}]], {n, 0, 10}]
%t A001008 m = 1; aa = {}; Do[k = 0; Do[k = k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, 
               Numerator[k]], {r, 1, 20}]; aa [From Artur Jasinski (grafix(AT)csl.pl), 
               Oct 16 2008]
%t A001008 Table[Numerator[Expand[EulerGamma/a + PolyGamma[0, 1 + a]/a]], {a, 1, 
               30}] [From Artur Jasinski (grafix(AT)csl.pl), Nov 02 2008]
%Y A001008 Cf. A002805, A007406, A007408, A007410, A075135.
%Y A001008 Cf. A001220(n) = Wieferich primes p: p^2 divides 2^(p-1) - 1. Cf. A125854, 
               A121999, A014566, A056903, A067657.
%Y A001008 A145609-A145640. [From Artur Jasinski (grafix(AT)csl.pl), Oct 16 2008]
%Y A001008 Sequence in context: A129082 A060746 A111935 this_sequence A096617 A025529 
               A124078
%Y A001008 Adjacent sequences: A001005 A001006 A001007 this_sequence A001009 A001010 
               A001011
%K A001008 nonn,easy,frac,nice
%O A001008 1,2
%A A001008 N. J. A. Sloane (njas(AT)research.att.com).

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