OFFSET
1,1
COMMENTS
NK(n,k) conjecture:
If k + 1 is prime then there are infinitely many primes of form:
NK(n,k) = n^k + (n+1)^k + (n+2)^k + ... + (n+k-1)^k + (n+k)^k
If k + 1 is not prime then gcd(NK(n,k), k + 1) > 1 with any positive integer n.
Some examples in the OEIS:
k = 1, primes of form NK(n,1) are all odd primes A065091.
k = 2, primes of form NK(n,2) is A027864.
k = 4, this sequence generates all primes of form NK(n,4).
All terms == 1 (mod 6). Bunyakovsky's conjecture implies that the sequence is infinite. - Robert Israel, Mar 29 2015
LINKS
Bui Quang Tuan, Table of n, a(n) for n = 1..1000
Wikipedia, Bunyakovsky conjecture
EXAMPLE
7 is in the sequence because 7^4 + 8^4 + 9^4 + 10^4 + 11^4 = 37699 which is prime.
MAPLE
F:= unapply(expand(add((n+i)^4, i=0..4)), n):
select(isprime, [seq(6*i+1, i=1..1000)]); # Robert Israel, Mar 29 2015
MATHEMATICA
Select[Range@ 2000, PrimeQ[#^4 + (# + 1)^4 + (# + 2)^4 + (# + 3)^4 + (# + 4)^4] &] (* Michael De Vlieger, Mar 26 2015 *)
Position[Partition[Range[2000]^4, 5, 1], _?(PrimeQ[Total[#]]&)]//Flatten (* Harvey P. Dale, Apr 28 2022 *)
PROG
(Magma) [n: n in [0..2*10^3] | IsPrime( n^4 + (n+1)^4 + (n+2)^4 + (n+3)^4 + (n+4)^4)]; // Vincenzo Librandi, Mar 27 2015
(Python)
from gmpy2 import is_prime
A256370_list = [n for n in range(1, 10**6) if is_prime(5*n*(n*(n*(n + 8) + 36) + 80) + 354)] # Chai Wah Wu, Mar 29 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Bui Quang Tuan, Mar 26 2015
STATUS
approved