--- In

primenumbers@yahoogroups.com,

"David Broadhurst" <d.broadhurst@...> wrote:

> "Numbers n such that there are no primes of the forms

> 2^m+3^n or 2^n+3^m for m < n"

These numbers include

1679, 1743, 4980, 4982, 5314, 5513, 5695, 6100, 7251, 8218

but my coverage of the range n < 8218 is not complete,

so this is not yet a bona fide sequence.

So far, I have tested more than 7000 values of n < 8300

and found these 10 examples.

This agrees tolerably well with my prior heuristics.

To make a rough estimate of the density of primes of the

form N = 2^a + 3^b, I sieved for prime factors < 10^5,

with a and b running from 5000 to 5200, and found that

5558 values of N survived. So I estimated the

probability of primality to be C/log(N), with

C =~ exp(Euler)*log(10^5)*5558/201^2 =~ 2.82.

Hence I guessed that the probability of finding that

a number n lies in this sequence exceeds

exp(-C/log(2))*exp(-C/log(3)) > 1/800.

I say "exceeds", since 2^n+3^m starts at size

O(2^n) and ends at size O(3^n), as m runs from

1 to n-1, while 2^m+3^n stays at size O(3^n).

David