Hi, Mark,

An interesting expression.

Let np ( n ) = n^n + (n+1)^(n+1) .

I confirm that n = 1, 2, and 3 are the only known primes of this form.

If n mod 6 = 4,

then 3 and ( n^2 + n + 1 ) / 3 (twice) are divisors.

If n = 415 , then np ( n ) has only 2 prime factors

and one of them is 29 .

You might want to look at posts on this subject to NMBRTHRY, e.g.,

http://listserv.nodak.edu/scripts/wa.exe?A2=ind0112&L=nmbrthry&P=R403
and

http://listserv.nodak.edu/scripts/wa.exe?A2=ind0209&L=nmbrthry&P=R1218
Also, you might want to try to fill in the gap left by the reply of

Ismael Jimenez Calvo in explaining the production of the interesting

factors cited.

There are some sequences in the On-line Encyclopedia. E.g.,

http://www.research.att.com/projects/OEIS?Anum=A056187.
Cheers.

Walter Nissen