A230444 Primes of the form (p^k + k - 1)/k for prime p and some k > 1.

5, 13, 61, 157, 181, 421, 601, 733, 821, 1741, 1861, 2287, 2521, 3121, 5101, 8581, 9661, 9931, 16381, 19609, 19801, 36721, 60901, 71821, 83641, 100801, 106261, 135721, 161881, 163021, 199081, 205441, 218461, 273061, 282001, 337021, 388081, 431521, 491041

Offset: 1

Author

Irina Gerasimova, Oct 18 2013

Keywords

• nonn

Examples

			601 is a term because (7^4 + 4 - 1)/4 = 601 where 7, 601 are both prime,
733 is a term because (13^3 + 3 -1)/3 = 733 where 13, 733 are both prime,
821 is a term because (3^8 + 8 - 1)/8 = 821 where 3, 821 are both prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A048161, A067756.

Programs

Maple

N:= 10^6: # for terms <= N
S:= {}: p:= 1:
do
  p:= nextprime(p);
  if p^2/2 > N then break fi;
  for k from 2 do
    v:= (p^k + k - 1)/k;
    if v > N then break fi;
    if v::integer and isprime(v) then  S:= S union {v} fi;
od od:
sort(convert(S,list)); # Robert Israel, Jun 22 2023

PARI

isA230444(n) = {isprime(n) || return(0); my(k = 2, v, p); while (1, v = k*n+1-k; if (ispower(v, k, &p) && isprime(p), return(1)); if (v < 2^k, return(0)); k++;);} \\ Michel Marcus, Oct 19 2013

Extensions

More terms from Michel Marcus, Oct 19 2013