# continuation of sequence

Edwin Clark eclark at math.usf.edu
Sat May 24 01:16:05 CEST 2003

```On Sat, 24 May 2003 all at abouthugo.de wrote:

>
> Joshua Zucker <Joshua_Zucker at castilleja.org> schrieb am 23.05.2003,
> 07:25:46:
> > 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
>
> Joshua, Neil, SeqFans,
>
> all the terms given above are prime. (Checked with
> Dario Alperns ECM Java Applet
> http://www.alpertron.com.ar/ECM.HTM ). I made some
> attempts up to n=638, without finding more terms,
> but I couldn't check the
> following list of numbers, most of them of the form
> 2*prime. Beginning with n=446=2*223=2*p,
> 2^n+(n/2)^n has more than 1000 decimal digits,
> exceeding my available prime testing tools'
> capabilities.
> 446,454,458,466,478,481,502,508,514,524,526,538,
> 542,548,554,556,562,566,568,579,584,586,596,603,
> 604,614,618,622,626,628,632,634
>
> Anyone out there to check primality of sum p_i^n
> for this list of numbers?
>
> Hugo Pfoertner

None of these work according to Maple's isprime -- which is a
probabilistic primality tester presumably correct when it returns false.

I was just checking up to 1000 with Maple and stopped my program to see
how far it got. It found no more values aside from 12, 88 and 207 for n
up to 822.

I'll let it run a little longer and let you know if I find any further
values of n that satisfy the condition.

--Edwin

```