1,2

H(n) is the maximal distance that a stack of n cards can project beyond the edge of a table without toppling.

By Wolstenholme's theorem, p^2 divides a(p-1) for prime p > 3.

Comments from Alexander Adamchuk (alex(AT)kolmogorov.com), Dec 11 2006 (Start)

p divides a(p^2-1) for prime p>3.

p divides a((p-1)/2) for prime p = {1093, 3511, ...} = A001220(n) = Wieferich primes p: p^2 divides 2^(p-1) - 1.

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].

a(n) is prime for n = {2, 3, 5, 8, 9, 21, 26, 41, 56, 62, 69, 79, 89, 91, ...} = A056903(n).

Corresponding primes a(n) are a(A056903(n)) = A067657(n) = {3, 11, 137, 761, 7129, 18858053, 34395742267, 85691034670497533, ...}. (End)

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]

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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Hongwei Chen, Evaluations of Some Variant Euler Sums, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.3.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 259.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, page 347.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 615.

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

R. M. Dickau, Harmonic numbers and the book-stacking problem

Hisanori Mishima, Factorizations of many number sequences

Hisanori Mishima, Factorizations of many number sequences

Hisanori Mishima, Factorizations of many number sequences

Hisanori Mishima, Factorizations of many number sequences

Hisanori Mishima, Factorizations of many number sequences

N. J. A. Sloane, Illustration of initial terms

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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

Eric Weisstein's World of Mathematics, Harmonic Mean

H(n) ~ log n + gamma + O(1/n) [see for example Hardy and Wright, Th. 422.]

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)

G.f. for H(n) : log(1-x)/(x-1) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 15 2003

H(n) = Sqrt[Sum[Sum[1/(i*j), {i, 1, n}], {j, 1, n}]] - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 24 2004

a(n)=Numerator[EulerGamma/n + PolyGamma[0, 1 + n]/n] [From Artur Jasinski (grafix(AT)csl.pl), Nov 02 2008]

H(n) = [ 1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520,... ].

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

a = 1; b = 1; maxN = 26; s = 0; Numerator[ Table[ s += 1/(a*n + b), {n, 0, maxN} ]]

H(n) = Table[Sqrt[Sum[Sum[1/(i*j), {i, 1, n}], {j, 1, n}]], {n, 0, 10}]

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]

Table[Numerator[Expand[EulerGamma/a + PolyGamma[0, 1 + a]/a]], {a, 1, 30}] [From Artur Jasinski (grafix(AT)csl.pl), Nov 02 2008]

Cf. A002805, A007406, A007408, A007410, A075135.

Cf. A001220(n) = Wieferich primes p: p^2 divides 2^(p-1) - 1. Cf. A125854, A121999, A014566, A056903, A067657.

A145609-A145640. [From Artur Jasinski (grafix(AT)csl.pl), Oct 16 2008]

Sequence in context: A129082 A060746 A111935 this_sequence A096617 A025529 A124078

Adjacent sequences: A001005 A001006 A001007 this_sequence A001009 A001010 A001011

nonn,easy,frac,nice

N. J. A. Sloane (njas(AT)research.att.com).