A243734 Primes p for which p + 4, p^2 + 4 and p^3 + 4 are primes.

3, 7, 103, 277, 487, 967, 4783, 5503, 5923, 8233, 21013, 26317, 27943, 41593, 55213, 78307, 78853, 86197, 89653, 94723, 99013, 123727, 148153, 157177, 166627, 172867, 177883, 179107, 185893, 192883, 194713, 203767, 204517, 223633, 225217, 227593, 236893

Offset: 1

Author

Abhiram R Devesh, Jun 09 2014

Keywords

• nonn
• easy

Comments

This is a subset of the sequences:

A023200: Primes p such that p + 4 is also prime.

A243583: Primes p for which p + 4 and p^3 + 4 are primes.

p is either 2 mod 5 or 3 mod 5, hence p^4 + 4 is 0 mod 5.

Examples

			p = 3 is in this sequence because p + 4 = 7, p^2 + 4 = 13 and p^3 + 4 = 31 are all primes.
p  : p+4,  p^2+4,     p^3+4
7  :  11,     53,       347
103: 107,  10613,   1092731
277: 281,  76733,  21253937
487: 491, 237173, 115501307
		

Links

• Abhiram R Devesh, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A023200, A243583.

Programs

PARI

s=[]; forprime(p=2, 200000, if(isprime(p+4) && isprime(p^2+4) && isprime(p^3+4), s=concat(s, p))); s \\ Colin Barker, Jun 11 2014

Python

import sympy.ntheory as snt
n=2
while n > 1 and n < 10**6:
    n1=n+4
    n2=((n**2)+4)
    n3=((n**3)+4)
    ##Check if n1, n2 and n3 are also primes.
    if snt.isprime(n1)== True and snt.isprime(n2)== True and snt.isprime(n3)== True:
        print(n, end=', ')
    n=snt.nextprime(n)