sequence (x+1)^x + x^x

[all at abouthugo.de]

cino hilliard <hillcino368 at hotmail.com> schrieb am 23.08.2003, 21:30:25:
> Hi Seq Fans,
> 
> a(n) =  (x+1)^x + x^x x=1, x=2,4,6,n for  n even
                       ^ is prime
> 
> I submitted this sequence having computed only 3 terms(contrary to the 
> rules)  thinking for sure it has
> more terms than 3,13,881.
> 

Cino,

you seem to have a lucky hand in finding
"nearly impossible to extend" sequences.
Finding primes of numbers growing like
x^x becomes extremely improbable, if you don't
have success for small x. This is a consequence
of the prime number theorem telling us that
the probability of finding a prime is ~1/log(x).

To give some examples (I tried to understand my
failure to find an extension of A097038
x^x+n is prime, x>1 for n=6,8,11,17,24,36;
nothing found until x=6000):

If we settle for only 10% probability to find
a prime after having found none up to x=1000,
then we need to continue to x=5048, or for 20%
probability up to x=48616.

If we have reached x=6000 without finding a prime
and have the computing
power to win the EFF 100000$ prize for finding
a 10^7-digit prime, (the current record
2^13466917-1 has 4053946 decimal digits) 
we can search up to x ~=1612000, but the
probability of success is still only ~0.22.

While I am writing this PFGW has reached x=2150
without finding new terms for your sequence:
(the 7000+ digits would not fit into an OEIS line
-anyway). It's likely to share the fate of
A087037 and A087038 (x^x+n is prime) ;-(

Hugo Pfoertner