OFFSET
1,2
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?
REFERENCES
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 156
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..500 from M. F. Hasler)
Eric Weisstein's World of Mathematics, Harmonic Number
Eric Weisstein's World of Mathematics, Continued Fraction
G. Xiao, Contfrac server, To evaluate H(m) and display its continued fraction expansion, operate on "sum(n=1, m, 1/n)"
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
EXAMPLE
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.
MATHEMATICA
Table[ Length[ ContinuedFraction[ HarmonicNumber[n]]], {n, 1, 75}] (* Robert G. Wilson v, Dec 22 2003 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Leroy Quet, Jul 10 2000
STATUS
approved