|
%I #39 Jan 23 2022 09:57:12
%S 4,31,227,1674,12367,91380,675214,4989191,36865412,272400600,
%T 2012783315,14872568831,109894245429,812014744422,6000022499693,
%U 44334502845080,327590128640500,2420581837980561,17885814992891026
%N Number of books required for n book-lengths of overhang in the harmonic book stacking problem. Sum_{i=1..a(n)} 1/i >= 2n and Sum_{i=1..a(n)-1} 1/i < 2n.
%C Bisection of A002387. - _Robert G. Wilson v_, Jan 24 2017
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 259.
%H Robert G. Wilson v, <a href="/A014537/b014537.txt">Table of n, a(n) for n = 1..1000</a> (terms 1..350 from Alois P. Heinz).
%H Mike Paterson and Uri Zwick, <a href="http://arXiv.org/abs/0710.2357">Overhang</a>, arXiv:0710.2357 [math.HO], 2007.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BookStackingProblem.html">Book Stacking Problem</a>
%F a(n) = A002387(2n), n>=1. Least a(n) with H(a(n)) > 2n with the harmonic numbers H(k):= A001008(k)/A002805(k).
%t f[n_] := (k = Floor[ N [ E^(n - EulerGamma) + 1/(2n), 24]] - 2; While[ Floor[ N[ Log[k] + EulerGamma + 1/(2k) - 1/(12k^2) + 1/(120k^4), 24]] < n, k++ ]; k); Table[ f[n], {n, 2, 32, 2} ]
%t a[n_] := k /. FindRoot[ HarmonicNumber[k] == 2*n, {k, Exp[2*n]}, WorkingPrecision -> 100] // Ceiling; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Jun 25 2013 *)
%Y Cf. A002387.
%K nonn,nice
%O 1,1
%A _Eric W. Weisstein_
%E More terms from _Robert G. Wilson v_, Dec 06 2001
%E Title corrected by _Jeremy Tan_, Sep 12 2020
|
