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A217734
Primes of the form x^2 + y^3 + 1 where x and y are prime.
2
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))
(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
KEYWORD
nonn
STATUS
approved