Product-Over-Integers Expansion Of Real

[Leroy Quet]

Hans Havermann wrote:
>> 2, 3, 23,...
>> (pi = 1/(1-1/2) *1/(1-1/3) *1/(1-1/23) *...,
>
>Perhaps someone can confirm these:
>
>2, 3, 23, 601, 1800857, 15150670259532, ...
>...

Hmmm... interesting.

I wonder somewhat too about the sequence restricted to the primes.

I specifically wonder about the primes (as opposed to, say, the squares, 
etc) because of the relationship between the product-expansion and the 
zeta-product.

Basically, if P is the set of primes whose product produces x

(x = product{p= elements of P} 1/(1-1/p) ),
and A is the set of all positive integers not divisible by any prime not 
in P 
(ie. A contains 1 and all positive integers divisible by only primes in 
P, and A contains only these integers), then 

x = sum{k= elements of A} 1/k.

And we could ask about the expansions which only involve primes raised to 
some fixed positive integer,

x = product{p=some of the primes} 1/(1- 1/p^n),

which converges for all n >= 2, obviously.

So, for x to have such an expansion, x must be <= zeta(n)
(and x must be >= 1).

thanks,
Leroy Quet