A091590 - OEIS

A091590

Number of terms in the simple continued fraction for the 10^n-th harmonic number, H_n = sum_{k=1 to n} (1/k).

1, 8, 68, 834, 8356, 84548, 841817, 8425934, 84277586

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OFFSET

0,2

COMMENTS

Conjecture: lim n -> infinity, a(n)/10^n -> C = 12*log(2)/Pi^2 = 0.842... - Benoit Cloitre, May 04 2002

REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 156.

LINKS

J. Sondow and E. W. Weisstein, MathWorld: Harmonic Number

FORMULA

a(n) = A055573(10^n). - Andrew Howroyd, Aug 10 2024

MATHEMATICA

s = 0; k = 1; Do[ While[s = s + 1/k; k < 10^n, k++ ]; Print[ Length[ ContinuedFraction[s]]]; k++, {n, 0, 6}]

Table[Length[ContinuedFraction[HarmonicNumber[10^n]]], {n, 0, 7}] (* Harvey P. Dale, Aug 24 2015 *)

CROSSREFS

Cf. A055573. n-th harmonic number H(m) = A001008(n)/A002805(n).

KEYWORD

hard,nonn

AUTHOR

EXTENSIONS

Corrected and extended by Eric W. Weisstein, Jan 23 2004

STATUS

approved