continuation of sequence

[Joshua Zucker]

OK, SOme quick hacking with mathematica extends the sequence at
http://primes.utm.edu/curios/page.php?short=539633

For the n, the base, I have
12, 88, 207
and for the prime "image" of n, sum of p^n over all prime factors p of n.
539633 = 2^12 + 2^12 + 3^12

43909277783870034878569768760415886733743786946105343887995366054267119200102384004474562849
= 2^88 + 2^88 + 2^88 + 11^88

7545048844883559926134754437031975993054726674236856637164268202580382602383411898370028958674896550375094045606979132820374507241375063026190221072559862927227552429910707339260761215523069351700952640157359769228765406964459882330366607234722984341
91295086941047447198115204429821 = 3^207 + 3^207 + 23^207

I haven't done any verification that these latter numbers are prime except
Mathematica's PrimeQ function.

And my algorithm was a VERY unsophisticated brute force search.  I'm sure
that anyone with a bit of intelligence can speed up the algorithm enough
to search some larger n.  I've checked up to n = 400 and found only these.

--Joshua Zucker
Castilleja School
joshua.zucker at stanfordalumni.org