A256370 - OEIS

%I #32 Sep 08 2022 08:46:11

%S 7,25,97,115,145,169,223,247,343,379,385,421,541,577,601,607,673,691,

%T 751,847,895,961,997,1111,1129,1237,1267,1303,1327,1459,1489,1555,

%U 1615,1639,1657,1663,1741,1765,1771,1807,1819,1831,1873,1903,1927,1945,1951,1963

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

%C NK(n,k) conjecture:

%C   If k + 1 is prime then there are infinitely many primes of form:

%C     NK(n,k) = n^k + (n+1)^k + (n+2)^k + ... + (n+k-1)^k + (n+k)^k

%C   If k + 1 is not prime then gcd(NK(n,k), k + 1) > 1 with any positive integer n.

%C Some examples in the OEIS:

%C   k = 1, primes of form NK(n,1) are all odd primes A065091.

%C   k = 2, primes of form NK(n,2) is A027864.

%C   k = 4, this sequence generates all primes of form NK(n,4).

%C All terms == 1 (mod 6). Bunyakovsky's conjecture implies that the sequence is infinite. - _Robert Israel_, Mar 29 2015

%H Bui Quang Tuan, <a href="/A256370/b256370.txt">Table of n, a(n) for n = 1..1000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bunyakovsky_conjecture">Bunyakovsky conjecture</a>

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

%p F:= unapply(expand(add((n+i)^4,i=0..4)), n):

%p select(isprime, [seq(6*i+1,i=1..1000)]); # _Robert Israel_, Mar 29 2015

%t Select[Range@ 2000, PrimeQ[#^4 + (# + 1)^4 + (# + 2)^4 + (# + 3)^4 + (# + 4)^4] &] (* _Michael De Vlieger_, Mar 26 2015 *)

%t Position[Partition[Range[2000]^4,5,1],_?(PrimeQ[Total[#]]&)]//Flatten (* _Harvey P. Dale_, Apr 28 2022 *)

%o (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

%o (Python)

%o from gmpy2 import is_prime

%o 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

%Y Cf. A027864.

%K nonn

%O 1,1

%A _Bui Quang Tuan_, Mar 26 2015