

A246519


Primes p such that 4+p, 4+p^2, 4+p^3 and 4+p^5 are all prime.


4



7, 5503, 21013, 301123, 303613, 420037, 469363, 679153, 771427, 991957, 999667, 1524763, 1707367, 2030653, 2333083, 2540563, 2552713, 2710933, 3009967, 3378103, 3441817, 3592213, 4419937, 4704613, 4840723, 5177797, 5691547, 6227587, 6275887, 6395677, 6595597, 6597163
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OFFSET

1,1


COMMENTS

For even k > 2, 4 + n^k is prime only for n = 1.
From Derek Orr, Aug 28 2014 (edited by Danny Rorabaugh, Apr 19 2015): (Start)
4+p^4 is composite for all primes p. For p = 2, 4+p^4 = 20 is composite. To prove it for odd primes, consider S(n) = 4+(2*n+1)^4. S(n) == 0 (mod 5) unless n == 2 (mod 5). If n == 2 (mod 5), then 2*n+1 == 0 (mod 5), which is only prime for n = 2; this gives p = 5 and 4+5^4 = 629 is composite. For other odd primes p, 4+p^4 is greater than 5 and divisible by 5.
4+p^(4*m) is also composite for any prime p and integer m > 0. For each m, the proof is the same as above.
(End)
All terms are == {3,7} (mod 10).  Zak Seidov, Aug 29 2014


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000


EXAMPLE

From K. D. Bajpai, Jan 20 2015: (Start)
a(2) = 5503:
4 + 5503 = 5507;
4 + 5503^2 = 30283013;
4 + 5503^3 = 166647398531;
4 + 5503^5 = 5046584669419727747;
all five are prime.
(End)


MATHEMATICA

k=4; Select[Prime[Range[1, 500000]], PrimeQ[k+#]&&PrimeQ[k+#^2] &&PrimeQ[k+#^3] &&PrimeQ[k+#^5]&] (*K. D. Bajpai, Jan 20 2015 *)


PROG

(PARI) for(n=1, 6000000, if(isprime(n) && isprime(4+n) && isprime(4+n^2) && isprime(4+n^3) && isprime(4+n^5), print1(n, ", "))) \\ Colin Barker, Aug 28 2014
(PARI) p=7; forprime(q=11, 1e8, if(qp==4 && isprime(4+p^2) && isprime(4+p^3) && isprime(4+p^5), print1(p, ", ")); p=q) \\ Charles R Greathouse IV, Aug 28 2014
(Python)
from sympy import prime, isprime
A246519_list = [p for p in (prime(n) for n in range(1, 10**5)) if all([isprime(4+p**z) for z in (1, 2, 3, 5)])]
# Chai Wah Wu, Sep 08 2014
(Magma) [p: p in PrimesUpTo(2*10^7)  IsPrime(4+p) and IsPrime(4+p^2) and IsPrime(4+p^3) and IsPrime(4+p^5)]; // Vincenzo Librandi, Apr 19 2015


CROSSREFS

Cf. A007591, A073573, A125260, A172367.
Primes p such that 4+p^7, 4+p^9 and 4+p^11 are also prime is A253937.  K. D. Bajpai, Jan 20 2015
The subsequence with 4+p^7 also prime is A246562.  Danny Rorabaugh, Apr 19 2015
Sequence in context: A095155 A219893 A243780 * A062644 A103174 A203693
Adjacent sequences: A246516 A246517 A246518 * A246520 A246521 A246522


KEYWORD

nonn


AUTHOR

Zak Seidov, Aug 28 2014


STATUS

approved



