[seqfan] Re: A079353

[David Applegate]

I suspect, without deep investigation, that what was meant by "best
rational approximation to" is "continued fraction convergent"

The continued fraction convergents to H(10)=7381/2520 are
2, 3, 41/14, 495/169, ...

The continued fraction convergents to H(31) are
4, 145/36, 149/37, 443/110, ...

The continued fraction convergents to H(82) are
4, 5, 499/100, 2001/401, ...

I haven't verified that the rest of the terms match this definition.

-Dave

> From seqfan-bounces at list.seqfan.eu Mon May 19 21:10:02 2014
> Date: 19 May 2014 18:09:31 -0700
> From: israel at math.ubc.ca
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] A079353

> The data in this sequence don't seem to fit the definition:

>          Numbers n such that the best rational approximation to H(n) with 
> denominator <=n is an integer, where H(n) denotes the n-th harmonic number

> The given data are 

>  	1, 3, 4, 10, 11, 30, 31, 82, 83, 226, 227, 615, 616, 1673, 1674

> For example, how does 10 fit in? H(10) = 7381/2520, and the best 
> approximation with denominator <= 10 is 29/10, which is not an integer. 
> Similarly, I don't see how 31, 82, 227, 616, or 1674 fit the definition, as 
> according to my computations the best approximations in these cases are 
> 125/31, 409/82, 1363/227, 4313/616, 13393/1674.

> Cheers,
> Robert

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