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

1,1

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.

Links

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

Example

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

Prog

(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)
(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

Crossrefs

Cf. A023200, A243583.

Sequence in context: A299377 A129660 A373806 * A372483 A158467 A260824

Adjacent sequences: A243731 A243732 A243733 * A243735 A243736 A243737

Keyword

nonn,easy

Author

Abhiram R Devesh, Jun 09 2014

Status

approved