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A256370 Positive integers n such that n^4 + (n+1)^4 + (n+2)^4 + (n+3)^4 + (n+4)^4 is prime. 1
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 (list; graph; refs; listen; history; text; internal format)
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
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
Cf. A027864.
Sequence in context: A146933 A155258 A261722 * A199116 A304421 A138729
KEYWORD
nonn
AUTHOR
Bui Quang Tuan, Mar 26 2015
STATUS
approved

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Last modified April 13 02:05 EDT 2024. Contains 371639 sequences. (Running on oeis4.)