A217734 Primes of the form x^2 + y^3 + 1 where x and y are prime.

13, 37, 53, 149, 151, 197, 317, 353, 389, 487, 557, 967, 1087, 1381, 1453, 1621, 1693, 1709, 1861, 1877, 2207, 2237, 2293, 2837, 3181, 3541, 3607, 3847, 4517, 4813, 5167, 5443, 5821, 6269, 6367, 6373, 6661, 6763, 6869, 6917, 7573, 7723, 7949, 8221, 9403, 9437

Offset

1,1

Comments

Terms are primes of the form 1 + A045699(n) for some n.

The first term that can be expressed two distinct ways is a(207) = 159079 = 101^2+53^3+1 = 367^2+29^3+1.

There are 27 values < 10^8 that can be expressed in two distinct ways.

Links

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Example

53 is in the sequence because 53 = 5^2 + 3^3 + 1 and 3, 5, and 53 are all prime numbers.

Prog

(PARI)
isa(n) = {
  my(r);
  forprime(x = 1, sqrtint(n),
      r = n - x ^ 2 - 1;
      if( ispower(r, 3, &p) && isprime(p),
          return(1)
      )
  );
  return(0)
}
select(isa, primes(1200))
\\ Felix Fröhlich & David A. Corneth & Peter Luschny, Jun 26 2021
(Python)
from sympy import isprime, primerange
def aupto(lim):
    sq = list(p**2 for p in primerange(1, int(lim**(1/2))+2))
    cb = list(p**3 for p in primerange(1, int(lim**(1/3))+2))
    s3 = set(s for s in (a + b + 1 for a in sq for b in cb) if s <= lim)
    return list(filter(isprime, sorted(s3)))
print(aupto(9999)) # Michael S. Branicky, Jun 24 2021
(SageMath)
def aList(sup):
    P = [p**2 for p in prime_range(1, int(sup**(1/2))+2)]
    Q = [(p**3 + 1) for p in prime_range(1, int(sup**(1/3))+2)]
    return sorted([a+b for a in P for b in Q if a+b <= sup and is_prime(a+b)])
print(aList(9999)) # Peter Luschny, Jun 25 2021

Crossrefs

Cf. A045700 (primes of the form p1^2 + p2^3).

Sequence in context: A343894 A155560 A353198 * A319969 A045809 A238675

Adjacent sequences: A217731 A217732 A217733 * A217735 A217736 A217737

Keyword

nonn

Author

Kevin L. Schwartz and Christian N. K. Anderson, Mar 22 2013

Status

approved