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A339692 Primes that can be expressed as p^k+2*k where p is prime and k >= 1.

%I A339692 #16 Dec 15 2020 18:29:38
%S A339692 5,7,13,19,29,31,43,53,61,73,89,103,109,131,139,151,173,181,193,199,
%T A339692 229,241,271,283,293,313,349,421,433,463,523,571,601,619,643,661,811,
%U A339692 823,829,859,883,1021,1033,1051,1063,1093,1153,1231,1279,1291,1303,1321,1373,1429,1453,1483,1489,1609
%N A339692 Primes that can be expressed as p^k+2*k where p is prime and k >= 1.
%C A339692 Terms expressible in more than one way include
%C A339692   13 = 11^1 + 2*1 = 3^2 + 2*2
%C A339692   349 = 347^1 + 2*1 = 7^3 + 2*3
%C A339692   78139 = 78137^^1 + 2*1 = 5^7 + 2*7
%C A339692   1092733 = 1092731^1 + 2*1 = 103^3 + 2*3
%C A339692   22665193 = 22665191^1 + 2*1 = 283^3 + 2*3.
%H A339692 Robert Israel, <a href="/A339692/b339692.txt">Table of n, a(n) for n = 1..10000</a>
%e A339692 a(5) = 29 is a term because 29 = 5^2 + 2*2. and 5 and 29 are primes.
%p A339692 N:= 1000: # for terms <= N
%p A339692 S:= {}:
%p A339692 for n from 1 while 3^n + 2*n <= N do
%p A339692   p:= 2:
%p A339692   do
%p A339692     p:= nextprime(p);
%p A339692     q:=  p^n + 2*n;
%p A339692     if q > N then break fi;
%p A339692     if isprime(q) then S:= S union {q};
%p A339692     fi
%p A339692 od od:
%p A339692 sort(convert(S,list));
%t A339692 Block[{nn = 1610, a = {}}, Do[Do[Which[# > nn, Break[], PrimeQ[#], AppendTo[a, #]] &[(#^k) + 2 k], {k, Infinity}] &[Prime@ i], {i, 2, PrimePi@ nn}]; Union@ a] (* _Michael De Vlieger_, Dec 13 2020 *)
%o A339692 (PARI) isok(p) = {if (isprime(p), for(k=1, p\2, if (k==isprimepower(p-2*k), return(1));););} \\ _Michel Marcus_, Dec 13 2020
%Y A339692 Includes A006512, A045637 and A201308.
%K A339692 nonn
%O A339692 1,1
%A A339692 _J. M. Bergot_ and _Robert Israel_, Dec 13 2020