A045700 Primes of form p^2+q^3 where p and q are primes.

17, 31, 347, 6863, 493043, 1092731, 1295033, 21253937, 22665191, 38272757, 54439943, 115501307, 904231067, 1121622323, 2738124203, 3067586681, 3301293173, 3673650011, 4549540397, 4599141251, 6507781367, 7222633241

Offset: 1

Author

Felice Russo

Keywords

• nice
• nonn
• easy

Comments

p and q cannot both be odd, thus p=2 or q=2. If q=2 then we want primes of form p^2+8. But 8=-1 mod 3. Since p is prime, p=3 or == 1 or 2 mod 3. If p=1 or 2 mod 3 then 3|p^2+8, so p=3. Therefore with the exception of the first entry (3^2+8=17) this sequence is really just primes of the form q^3+4.

Examples

			a(4) = 6863 = 19^3 + 2^2.
		

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A045699.

Programs

Maple

for n from 1 to 1000 do if (isprime((ithprime(n))^3+4)) then print((ithprime(n))^3+4,4); fi; if (isprime((ithprime(n))^2+8)) then print((ithprime(n))^2+8,8); fi; od;

Mathematica

Join[{17},Select[Prime[Range[300]]^3+4,PrimeQ]] (* Harvey P. Dale, Jul 20 2011 *)

PARI

list(lim)=my(v=List([17]), t); lim\=1; forprime(p=3,sqrtnint(lim\1-4,3), if(isprime(t=p^3+4), listput(v, t))); Set(v) \\ Charles R Greathouse IV, Feb 07 2017

Formula

Primes in A045699.

Extensions

Extension and comment from Joe DeMaio (jdemaio(AT)kennesaw.edu)