cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A055573 Number of terms in simple continued fraction for n-th harmonic number H_n = Sum_{k=1..n} (1/k).

Original entry on oeis.org

1, 2, 3, 2, 5, 4, 6, 7, 10, 8, 7, 10, 15, 9, 9, 17, 18, 11, 20, 16, 18, 18, 23, 19, 24, 25, 24, 26, 29, 21, 24, 23, 26, 25, 32, 34, 33, 26, 24, 31, 32, 31, 36, 36, 39, 32, 34, 42, 47, 44, 46, 35, 40, 48, 43, 47, 59, 50, 49, 39, 50, 66, 54, 44, 54, 49, 41, 64, 47, 46, 54, 71, 72
Offset: 1

Views

Author

Leroy Quet, Jul 10 2000

Keywords

Comments

By "simple continued fraction" is meant a continued fraction whose terms are positive integers and the final term is >= 2.
Does any number appear infinitely often in this sequence?

Examples

			Sum_{k=1 to 3} [1/k] = 11/6 = 1 + 1/(1 + 1/5), so the 3rd term is 3 because the simple continued fraction for the 3rd harmonic number has 3 terms.
		

References

  • Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 156

Crossrefs

m-th harmonic number H(m) = A001008(m)/A002805(m).
Cf. A139001 (partial sums).

Programs

  • Mathematica
    Table[ Length[ ContinuedFraction[ HarmonicNumber[n]]], {n, 1, 75}] (* Robert G. Wilson v, Dec 22 2003 *)
  • PARI
    c=0;h=0;for(n=1,500,write("projects/b055573.txt",c++," ",#contfrac(h+=1/n))) \\ M. F. Hasler, May 31 2008
    
  • Python
    from sympy import harmonic
    from sympy.ntheory.continued_fraction import continued_fraction
    def A055573(n): return len(continued_fraction(harmonic(n))) # Chai Wah Wu, Jun 27 2024

Formula

It appears that lim n -> infinity a(n)/n = C = 0.84... - Benoit Cloitre, May 04 2002
Conjecture: limit n -> infinity a(n)/n = 12*log(2)/Pi^2 = 0.84..... = A089729 Levy's constant. - Benoit Cloitre, Jan 17 2004