A256370 Positive integers n such that n^4 + (n+1)^4 + (n+2)^4 + (n+3)^4 + (n+4)^4 is prime.

7, 25, 97, 115, 145, 169, 223, 247, 343, 379, 385, 421, 541, 577, 601, 607, 673, 691, 751, 847, 895, 961, 997, 1111, 1129, 1237, 1267, 1303, 1327, 1459, 1489, 1555, 1615, 1639, 1657, 1663, 1741, 1765, 1771, 1807, 1819, 1831, 1873, 1903, 1927, 1945, 1951, 1963

Offset: 1

Author

Bui Quang Tuan, Mar 26 2015

Keywords

• nonn

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

Examples

			7 is in the sequence because 7^4 + 8^4 + 9^4 + 10^4 + 11^4 = 37699 which is prime.
		

Links

• Bui Quang Tuan, Table of n, a(n) for n = 1..1000
• Wikipedia, Bunyakovsky conjecture

Crossrefs

Cf. A027864.

Programs

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

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 *)

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