A122395 Primes of the form p^k - p^(k-1) - 1, with p prime and k>1.

3, 5, 7, 17, 19, 31, 41, 53, 109, 127, 271, 293, 499, 811, 929, 2027, 2161, 3659, 4373, 4421, 4969, 8191, 9311, 10099, 13121, 13309, 16001, 17029, 19181, 22051, 32579, 38611, 57839, 72091, 78607, 93941, 109229, 128521, 131071, 143261, 157211

Offset: 1

Author

T. D. Noe, Aug 31 2006

Keywords

• nonn

Comments

The paper by Stein and Williams gives a method for finding primes of this form when k>(p+1)/2.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Andreas Stein and H. C. Williams, Explicit primality criteria for (p-1)p^n-1, Math. Comp. 69 (2000), 1721-1734.

Crossrefs

Cf. A122396.

Programs

Maple

N:= 10^6: # for terms <= N
p:= 1: R:= NULL:
do
  p:= nextprime(p);
  if p^2 - p - 1 > N then break fi;
  for k from 2 do
    q:= p^k - p^(k-1)-1;
    if q > N then break fi;
    if isprime(q) then R:= R, q fi;
od od:
sort(convert({R},list)); # Robert Israel, Mar 12 2023

Mathematica

nn=10^6; lst={}; n=1; While[p=Prime[n]; k=2; While[m=p^k-p^(k-1)-1; m2, n++ ]; lst=Union[lst]