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

B.W.J. Irwin bwji2 at cam.ac.uk
Wed Dec 21 18:34:01 CET 2016


I was prompted by Joerg to ask here: (I've just joined so apologies if 
this is the wrong format)

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.

Thanks,

Benedict Irwin


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