A113123 Numerator of next-best approximation to harmonic numbers. a(n) = Numerator of (A055573(n)-1)th convergent of n-th harmonic number, Sum_{k=1..n} 1/k.

0, 1, 2, 2, 16, 22, 70, 106, 1836, 2639, 14281, 21167, 167857, 87932, 169452, 923889, 3590229, 950596, 40366604, 23213361, 517630, 1391957, 160363133, 222528683, 10125035246, 4324958013, 81828906108, 71315450571, 4320297286472

Offset

1,3

Comments

A100398 gives terms of continued fractions of harmonic numbers.

For n >= 2, a(n) = the denominator of the ratio equal to the continued fraction made by reversing the order of the terms of the continued fraction of the n-th harmonic number. (The numerator of this ratio is the numerator of the n-th harmonic number, A001008(n).) - Leroy Quet, Dec 24 2006

Links

Table of n, a(n) for n=1..29.

Example

H(6) = 49/20 = 2 +1/(2 +1/(4 +1/2)), so a(6) = numerator of 2 +1/(2 +1/4) = 22/9.

Prog

(PLT Scheme) ;; (harmonic n) is the n-th harmonic sum
;; frac->cf and cf->frac are utility functions that convert fractions to continued fractions and vice versa.
(define (A113123 n)
(cond
[(= n 1) 0]
[else (numerator (cf->frac (reverse (rest (reverse (frac->cf (harmonic n)))))))]))
;; Joshua Zucker, May 08 2006

Crossrefs

Cf. A100398, A055573, A113124.

Sequence in context: A240033 A088139 A152556 * A353912 A303567 A177832

Adjacent sequences: A113120 A113121 A113122 * A113124 A113125 A113126

Keyword

easy,frac,nonn

Author

Leroy Quet, Oct 14 2005

Extensions

More terms from Joshua Zucker, May 08 2006

Status

approved