[seqfan] Re: Engel Expansion of constants = p-rough numbers, prime p

[Olivier Gerard]

On Wed, Dec 21, 2016 at 6:34 PM, B.W.J. Irwin <bwji2 at cam.ac.uk> wrote:

> I was prompted by Joerg to ask here: (I've just joined so apologies if
> this is the wrong format)
>
> Thanks Joerg for that. there are only a few rules and you respected two
important ones : you wrote in plain text
and you sent links instead of including long material.


> Key points:
> Engel Expansion: A unique weakly increasing integer sequence corresponding
> to a real constant.
> http://mathworld.wolfram.com/EngelExpansion.html
>
> For most well behaved sequences, there seems to be a usually complicated
> constant than then contains the information of the sequence.
>
> p-rough numbers: The set of integers not divisible by primes less than p.
> http://mathworld.wolfram.com/RoughNumber.html
>
> For larger p, the sequence gets denser in primes as the small divisors are
> sifted out.
> In general for p-rough numbers, all the numbers between a(1)=1 to
> a(...)=p^2 are prime.
>
> I think I have found a general expression (infinite sum, or finite sum of
> hypergeometric functions) for the constant whose Engel expansion is the
> p-rough numbers, for any p. The higher terms are just conjectural at the
> moment, apart form the 5-rough and 7-rough numbers, but they seem to work.
> The constant seems to tend to 1 as p tends to infinity.
>
> https://www.authorea.com/users/5445/articles/144462/_show_article
>
> They get quickly harder to evaluate as p increases, but seem to be working
> for the first few.
>
> Mainly pertains to A279664.
>
> Any comments/ideas/criticisms are welcome.
>

I like the last formula (18) very much but I think the second k in the
denominator of the first fraction should be in the argument of the second
gamma function.

With my best regards,

Olivier Gérard