PartitionsQ / A046065 extended

[Dean Hickerson]

Mostly to Eric W. Weisstein

> I have found the next largest member of A046065 (number of partitions of n
> into distinct parts (A000009[ n ]) is prime)
>
> 5, 7, 22, 70, 100, 495, 1247, 2072, 320397
...
> I was surprised to find 320397, and it seems _very_ unlikely (i.e.,
> exponentially small chance) that any more exist.

Why?

Asymptotically,

                 1         pi sqrt(m/3)
    q(m) ~ -------------  e            .
               1/4   3/4
           4  3     m

q(m) is only odd if m is pentagonal, i.e.  m = n(3n+1)/2  for some
integer n (positive, negative, or zero).  Substituting in the above,

      n(3n+1)         -3/2  B |n|
    q(-------) ~ A |n|     e
         2

for positive constants A and B.  The probability that a "random"
integer N is prime is about  1/log(N);  for  N = q(n(3n+1)/2) 
this is about  1/(B |n|).  Since the harmonic series diverges,
I'd expect there to be infinitely many primes of this type.

Do you know of divisibility properties of q(n) which reduce the
probability of primality, to make the expected number of primes
finite?

Dean Hickerson
dean at math.ucdavis.edu