[seqfan] Engel Expansion of constants = p-rough numbers, prime p
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)
Engel Expansion: A unique weakly increasing integer sequence
corresponding to a real constant.
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
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.
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.
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