A121710 The smallest prime of the form (prime(n+1)^k + prime(n+2)^k)/2 for positive integer k.

17, 37, 8521, 21601, 229, 106921, 205081, 289278699121, 815401, 1398841, 8274567108488469403564696641244659777685186165444353190460129729940809291805549571887038803603334751361, 3122281, 2029

Offset: 1

Author

Cino Hilliard, Sep 10 2006

Keywords

• nonn

Comments

These numbers are all of the form 4n+1.

k needs to be a power of two. The sequence of the associated k is 2, 2, 4, 4, 2, 4, 4, 8, 4, 4, 64, 4, 2, 0, 32, 4, 4, 4, 0, 0, 0, 8, 0, 0, 8, 4, 8, 4, 2, 4, 0, 32, 0, 2, 8, 2, ... where 0 is inserted if a(n) does not appear to exist. - Robert G. Wilson v, Aug 02 2018

It seems likely that a(14) does not exist. No k <= 2^15 works. - Don Reble and Robert Israel, Aug 02 2018

Links

• Zak Seidov, query about (3^n + 5^n)/2 is prime in seqfan 9/10/06.

Programs

Maple

A121710 := proc(n)
    local p1,p2,k,a ;
    p1 := ithprime(n+1) ;
    p2 := nextprime(p1) ;
    for k from 1 do
        a := (p1^k+p2^k)/2 ;
        if type(a,'integer') and isprime(a) then
            return a;
        end if;
    end do:
end proc:
for n from 1 do
    printf("%d %d\n",n,A121710(n)) ;
end do: # R. J. Mathar, Aug 02 2018

PARI

g(n,a,b) = for(x=1,n,y=(a^x+b^x)/2;if(ispseudoprime(y),print(a","b","x","y)))

Extensions

Name and Data corrected by Robert Israel, Aug 02 2018