A339692 - OEIS

%I #16 Dec 15 2020 18:29:38

%S 5,7,13,19,29,31,43,53,61,73,89,103,109,131,139,151,173,181,193,199,

%T 229,241,271,283,293,313,349,421,433,463,523,571,601,619,643,661,811,

%U 823,829,859,883,1021,1033,1051,1063,1093,1153,1231,1279,1291,1303,1321,1373,1429,1453,1483,1489,1609

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

%C Terms expressible in more than one way include

%C   13 = 11^1 + 2*1 = 3^2 + 2*2

%C   349 = 347^1 + 2*1 = 7^3 + 2*3

%C   78139 = 78137^^1 + 2*1 = 5^7 + 2*7

%C   1092733 = 1092731^1 + 2*1 = 103^3 + 2*3

%C   22665193 = 22665191^1 + 2*1 = 283^3 + 2*3.

%H Robert Israel, <a href="/A339692/b339692.txt">Table of n, a(n) for n = 1..10000</a>

%e a(5) = 29 is a term because 29 = 5^2 + 2*2. and 5 and 29 are primes.

%p N:= 1000: # for terms <= N

%p S:= {}:

%p for n from 1 while 3^n + 2*n <= N do

%p   p:= 2:

%p   do

%p     p:= nextprime(p);

%p     q:=  p^n + 2*n;

%p     if q > N then break fi;

%p     if isprime(q) then S:= S union {q};

%p     fi

%p od od:

%p sort(convert(S,list));

%t 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 (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 Includes A006512, A045637 and A201308.

%K nonn

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Dec 13 2020