A222119 Number k yielding the smallest prime of the form (k+1)^p - k^p, where p = prime(n).

1, 1, 1, 1, 5, 1, 1, 1, 5, 2, 1, 39, 6, 4, 12, 2, 2, 1, 6, 17, 46, 7, 5, 1, 25, 2, 41, 1, 12, 7, 1, 7, 327, 7, 8, 44, 26, 12, 75, 14, 51, 110, 4, 14, 49, 286, 15, 4, 39, 22, 109, 367, 22, 67, 27, 95, 80, 149, 2, 142, 3, 11, 402, 3, 44, 10, 82, 20, 95, 4, 108, 349, 127, 303, 37, 3, 162

Offset: 1

Author

Vladimir Pletser, Feb 07 2013

Keywords

• nonn

Comments

The smallest k generating a prime of the form (k+1)^p - k^p (A121620) for the prime A000040(n). For the primes p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, ... (A000043), k = 1 and Mersenne primes 2^p - 1 (A000668) are obtained. For p = 11, 23, 29, ..., the smallest primes of the form (k+1)^p - k^p are respectively 313968931 (for k = 5), 777809294098524691 (for k = 5 also), 68629840493971 (for k = 2), ..., so a(5) = 5, a(9) = 5, a(10) = 2, ...

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..175

Crossrefs

Cf. A103794, A222120 (number of digits in the primes).

Programs

Maple

A222119 := proc(n)
        p := ithprime(n) ;
        for k from 1 do
                if isprime((k+1)^p-k^p) then
                        return k;
                end if;
        end do:
end proc: # R. J. Mathar, Feb 10 2013

Mathematica

Table[p = Prime[n]; k = 1; While[q = (k + 1)^p - k^p; ! PrimeQ[q], k++]; k, {n, 80}] (* T. D. Noe, Feb 12 2013 *)

PARI

f(p) = {my(k=1); while(ispseudoprime((k+1)^p-k^p)==0, k++); k; }
lista(nn) = forprime(p=2, nn, print1(f(p), ", ")); \\ Jinyuan Wang, Feb 03 2020

Formula

a(n) = A103794(n) - 1. - Ray Chandler, Feb 26 2017

Extensions

More terms from Ray Chandler, Feb 27 2017