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

5, 7, 13, 19, 29, 31, 43, 53, 61, 73, 89, 103, 109, 131, 139, 151, 173, 181, 193, 199, 229, 241, 271, 283, 293, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1373, 1429, 1453, 1483, 1489, 1609

Offset: 1

Author

J. M. Bergot and Robert Israel, Dec 13 2020

Keywords

• nonn

Comments

Terms expressible in more than one way include

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

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

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

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

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

Examples

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

Links

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

Crossrefs

Includes A006512, A045637 and A201308.

Programs

Maple

N:= 1000: # for terms <= N
S:= {}:
for n from 1 while 3^n + 2*n <= N do
  p:= 2:
  do
    p:= nextprime(p);
    q:=  p^n + 2*n;
    if q > N then break fi;
    if isprime(q) then S:= S union {q};
    fi
od od:
sort(convert(S,list));

Mathematica

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 *)

PARI

isok(p) = {if (isprime(p), for(k=1, p\2, if (k==isprimepower(p-2*k), return(1));););} \\ Michel Marcus, Dec 13 2020