Primes sorted by relation of largest divisors of p+-1

[Dean Hickerson]

Hugo Pfoertner wrote:

> BTW, equal number of divisors is
> http://www.research.att.com/projects/OEIS?Anum=A067889

For reference, A067889 is defined as "Primes p such that tau(p+1)=tau(p-1)
where tau(k)=A000005(k)".

> Is there any explanation for Benoit Cloitre's observation for this
> sequence:
> 
> a(n) seems curiously to be asymptotic to 25*n*Log(n) ?

Numerically, some such statement seems reasonable (for some value of 25):  For
n from 1000 up to 100000,  a(n)/(n log(n))  stays between 25.17 and 26.12, and
it looks like it might be converging slowly to some limit.

But heuristically, this is implausible.  The n-th prime is asymptotic to
n log(n),  so if  a(n)/(n log(n))  approaches a finite limit L, then
the probability that  tau(p-1)=tau(p+1),  for a random prime p, is
1/L > 0.  I can't prove that that's false, but for a function that varies
as erratically as tau(n), I'm sure that the probability must be 0.

Perhaps someone who knows more about the distribution of tau(n) could suggest
a more likely asymptotic formula for a(n).

Dean Hickerson
dean at math.ucdavis.edu