A286658 Primes of the form p*b^b + 1, where p is a prime and b>1.

13, 29, 53, 149, 173, 269, 293, 317, 389, 509, 557, 653, 769, 773, 797, 1109, 1229, 1493, 1637, 1733, 1949, 1997, 2309, 2477, 2693, 2837, 2909, 2957, 3329, 3413, 3533, 3677, 3989, 4133, 4157, 4253, 4349, 4373, 4493, 4517, 5189, 5309, 5693, 5717, 5813, 6173

Offset: 1

Author

Vincenzo Librandi, May 12 2017

Keywords

• nonn
• easy

Examples

			a(1) = 3*(2^2)+1 = 13.
a(2) = 7*(2^2)+1 = 29.
a(3) = 13*(2^2)+1 = 53.
		

Links

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

Crossrefs

Cf. A175768, A285015.

Programs

Maple

N:= 10000: # for all terms <= N
Res:= NULL:
P:= select(isprime, [2,seq(i,i=3..N/4,2)]):
for b from 2  do
  q:= b^b; if q > N/2 then break fi;
  for i from 1 to nops(P) do
     x:= P[i]*q+1;
     if x > N then break fi;
     if isprime(x) then Res:= Res, x fi;
od od:
sort(convert({Res},list)); # Robert Israel, Nov 12 2019

Mathematica

nmax=10^4; pimax=PrimePi[nmax]; bmax=1;While[(bmax+1)^(bmax+1)<=nmax,bmax++];Select[Union@Flatten@Table[Prime[pi] b^b+1,{b,2,bmax},{pi,pimax}],PrimeQ[#]&&#<=nmax&]

PARI

list(lim)=my(v=List()); lim\=1; for(b=2,oo, my(p=2*b^b+1); if(p>lim, break); if(isprime(p), listput(v,p))); forstep(b=2,oo,2, my(B=b^b); if(3*B+1>lim, break); forprime(q=3,(lim-1)\B, my(p=q*B+1); if(isprime(p), listput(v,p)))); Set(v) \\ Charles R Greathouse IV, Jun 16 2022