A272032 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.

7, 13, 43, 127, 281

Offset

1,1

Comments

All terms are prime (conjecture).

The variables q and r are the number of composite numbers between the previous and following successive primes of the number n.

They seem to be extremely rare. I checked them for {q,r}<=15 and n < 10^5.

Numbers n with q + r = 0 lead to 0/0 and are excluded. - Wolfdieter Lang, Apr 22 2016

Links

Table of n, a(n) for n=1..5.

Example

For a(1)=7, q=1 and r=3, (1^7+3^7)/(1+3)=547, 547 is prime.

Mathematica

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 *)

Prog

(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
(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

Crossrefs

Cf. A046933.

Sequence in context: A047977 A139403 A087820 * A023286 A287685 A159305

Adjacent sequences: A272029 A272030 A272031 * A272033 A272034 A272035

Keyword

nonn,more

Author

Marc Morgenegg, Apr 18 2016

Extensions

a(1) added by Charles R Greathouse IV, Apr 18 2016

Status

approved