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

%I #25 Jun 23 2023 16:56:27

%S 5,13,61,157,181,421,601,733,821,1741,1861,2287,2521,3121,5101,8581,

%T 9661,9931,16381,19609,19801,36721,60901,71821,83641,100801,106261,

%U 135721,161881,163021,199081,205441,218461,273061,282001,337021,388081,431521,491041

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

%H Robert Israel, <a href="/A230444/b230444.txt">Table of n, a(n) for n = 1..10000</a>

%e 601 is a term because (7^4 + 4 - 1)/4 = 601 where 7, 601 are both prime,

%e 733 is a term because (13^3 + 3 -1)/3 = 733 where 13, 733 are both prime,

%e 821 is a term because (3^8 + 8 - 1)/8 = 821 where 3, 821 are both prime.

%p N:= 10^6: # for terms <= N

%p S:= {}: p:= 1:

%p do

%p p:= nextprime(p);

%p if p^2/2 > N then break fi;

%p for k from 2 do

%p v:= (p^k + k - 1)/k;

%p if v > N then break fi;

%p if v::integer and isprime(v) then S:= S union {v} fi;

%p od od:

%p sort(convert(S,list)); # _Robert Israel_, Jun 22 2023

%o (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

%Y Cf. A048161, A067756.

%K nonn

%O 1,1

%A _Irina Gerasimova_, Oct 18 2013

%E More terms from _Michel Marcus_, Oct 19 2013