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Search: id:A055573
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| A055573 |
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Number of terms in simple continued fraction for n-th harmonic number H_n = sum_{k=1 to n} [1/k]. |
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+0
19
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| 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)
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OFFSET
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1,2
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COMMENT
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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?
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 156
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LINKS
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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)"
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FORMULA
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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
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EXAMPLE
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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.
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MATHEMATICA
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Table[ Length[ ContinuedFraction[ HarmonicNumber[n]]], {n, 1, 75}] (from Robert G. Wilson v Dec 22 2003)
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PROGRAM
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(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
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Leroy Quet Jul 10 2000
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