Can we count the no. of $x$ where $ p^{\alpha -1} < x < p^{\alpha}$ , $gcd(x, 2p)=1$ and if $d |x$ and $d < p ^{\beta}$ for some $1< \beta<\alpha-1$ then $ \frac {x} {d} > p^{\alpha - \beta}$ .

Why do you want to do this?
– TonyKNov 7 '11 at 10:06

it helps me to find primes in doubly stochastic matrices.
– KamalakshyaNov 7 '11 at 10:09

Oh? The entries in stochastic matrices are between zero and one - there are no primes in stochastic matrices. I wonder what you mean.
– Gerry MyersonNov 7 '11 at 11:17

4

doubly stochastic matrices forms a semigroup. So one can talk about irreducible elements, i call them primes. I am looking at the doubly stochastic matrices whose entries are rational. If i cleared out the denominators, these matrices will give rise to matrices with row and column sum constant, and i will take care that the gcd of all it's entries to be 1. The above problem with some more data counts how many 2*2 prime matrices are there with row and column sum $p^{\alpha}$.
– KamalakshyaNov 7 '11 at 15:14