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%I #22 Jun 27 2016 16:15:57
%S 7,13,43,127,281
%N Positive integers n such that (q^n + r^n)/(q+r) is prime, where q is the number of consecutive composite numbers smaller than n and r is the number of consecutive composite numbers greater than n.
%C All terms are prime (conjecture).
%C The variables q and r are the number of composite numbers between the previous and following successive primes of the number n.
%C They seem to be extremely rare. I checked them for {q,r}<=15 and n < 10^5.
%C Numbers n with q + r = 0 lead to 0/0 and are excluded. - _Wolfdieter Lang_, Apr 22 2016
%e For a(1)=7, q=1 and r=3, (1^7+3^7)/(1+3)=547, 547 is prime.
%t Select[Range[3, 10^3], Function[n, With[{q = n - NextPrime[n, -1] - 1, r = NextPrime@ n - n - 1}, If[q + r == 0, False, PrimeQ[(q^n + r^n)/(q + r)]]]]] (* _Michael De Vlieger_, Apr 18 2016 *)
%o (PARI) is(n)=my(q=n-precprime(n-1)-1,r=nextprime(n+1)-n-1,t); q+r && denominator(t=(q^n + r^n)/(q+r))==1 && isprime(t) \\ _Charles R Greathouse IV_, Apr 18 2016
%o (PARI) list(lim)=my(v=List(),p=2,q=3,t); forprime(r=5,nextprime(lim+1), t=((q-p-1)^q+(r-q-1)^q)/(r-p-2); if(denominator(t)==1 && ispseudoprime(t), listput(v,q)); p=q; q=r); Vec(v) \\ _Charles R Greathouse IV_, Apr 18 2016
%Y Cf. A046933.
%K nonn,more
%O 1,1
%A _Marc Morgenegg_, Apr 18 2016
%E a(1) added by _Charles R Greathouse IV_, Apr 18 2016
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