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A055573 Number of terms in simple continued fraction for n-th harmonic number H_n = sum_{k=1 to n} [1/k]. +0
19
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 (list; graph; listen)
OFFSET

1,2

COMMENT

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

M. F. Hasler, Table of n, a(n) for n=1,...,500.

Leroy Quet, Home Page (listed in lieu of email address)

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 (benoit7848c(AT)orange.fr), May 04 2002

Conjecture : limit n ->infty a(n)/n = 12*ln(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 3_rd term is 3 because the simple continued fraction for the 3_rd harmonic number has 3 terms.

MATHEMATICA

Table[ Length[ ContinuedFraction[ HarmonicNumber[n]]], {n, 1, 75}] (from Robert G. Wilson v Dec 22 2003)

PROGRAM

(PARI) c=0; h=0; for(n=1, 500, write("projects/b055573.txt", c++, " ", #contfrac(h+=1/n))) - M. F. Hasler (www.univ-ag.fr/~mhasler), May 31 2008

CROSSREFS

m-th harmonic number H(m) = A001008(m)/A002805(m).

Cf. A058027, A100398, A110020, A112286, A112287.

Cf. A139001 (partial sums).

Sequence in context: A089587 A067316 A127433 this_sequence A072969 A139712 A075365

Adjacent sequences: A055570 A055571 A055572 this_sequence A055574 A055575 A055576

KEYWORD

nonn

AUTHOR

Leroy Quet Jul 10 2000

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Last modified November 25 20:09 EST 2009. Contains 167514 sequences.


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