OFFSET
1,1
COMMENTS
These primes are important in studying k-imperfect numbers (A127724), see Iannucci-link. Except for the cases p^e = 3 and 8, which yield primes 2 and 5, e is an even number such that e+1 is prime. In fact, except for those two cases, all the primes are of the form (1+p^q)/(1+p), where q is an odd prime; that is, repunit primes with negative prime base.
LINKS
David A. Corneth, Table of n, a(n) for n = 1..33914 (terms < 10^14) (the first 4799 terms < 10^12 from T. D. Noe)
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), Article 00.2.7.
Douglas E. Iannucci, On a variation of perfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 6 (2006), #A41.
EXAMPLE
From David A. Corneth, Oct 28 2017: (Start)
For (p, e) = (3, 1) we have the prime 3^1 - 3^0 = 2.
For (p, e) = (2, 3) we have the prime 2^3 - 2^2 + 2^1 - 2^0 = 5.
The examples above are the cases mentioned in the comments not of the form (1+p^q)/(1+p). A prime of that form is below;
For (p, e) = (2, 4) we have the prime 2^4 - 2^3 + 2^2 - 2^1 + 2^0 = 11 = (1+p^(e+1)) / (1+p) = 33/3.
PROG
(PARI) upto(n) = {my(res = List([2, 5])); forprime(p = 2, sqrtnint(n, 2), forprime(q = 3, logint(n * (1+p), p), r = (1+p^q)/(1+p); if(isprime(r), listput(res, r)))); listsort(res, 1); res} \\ David A. Corneth, Oct 28 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
T. D. Noe, Jan 25 2007
STATUS
approved