A014537 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.

4, 31, 227, 1674, 12367, 91380, 675214, 4989191, 36865412, 272400600, 2012783315, 14872568831, 109894245429, 812014744422, 6000022499693, 44334502845080, 327590128640500, 2420581837980561, 17885814992891026

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• nice

Comments

Bisection of A002387. - Robert G. Wilson v, Jan 24 2017

References

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

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1000 (terms 1..350 from Alois P. Heinz).
• Mike Paterson and Uri Zwick, Overhang, arXiv:0710.2357 [math.HO], 2007.
• Eric Weisstein's World of Mathematics, Book Stacking Problem

Crossrefs

Cf. A002387.

Programs

Mathematica

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} ]
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 *)

Formula

a(n) = A002387(2n), n>=1. Least a(n) with H(a(n)) > 2n with the harmonic numbers H(k):= A001008(k)/A002805(k).

Extensions

More terms from Robert G. Wilson v, Dec 06 2001

Title corrected by Jeremy Tan, Sep 12 2020