OFFSET
1,1
COMMENTS
For exponent 2 instead of 4 see A089489: Pythagorean triple has a prime hypotenuse.
Corresponding sequences with odd exponent u are impossible: x^u + y^u has factor x+y.
a(2k-1) is even, a(2k) is odd, a(n)-n is odd.
Conjecture: a(n) exists for all n, i.e., the sequence is well-defined and infinite.
Conjecture: a(n)-n = 1 for infinitely many n.
The largest value of a(n)-n for n <= 100 occurs at n = 90: 121-90 = 31.
a(n)-n = 1 for 35 values of n <= 100.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
EXAMPLE
1^4 + 2^4 = 17 is prime, so a(1) = 2.
2^4 + 3^4 = 97 is prime, so a(2) = 3.
5^4 + 6^4 = 1921 = 17*113, 5^4 + 7^4 = 3026 = 2*17*89, 5^4 + 8^4 = 4721 is prime, so a(5) = 8.
MATHEMATICA
sn[n_]:=Module[{k=n+1, n4=n^4}, While[CompositeQ[n4+k^4], k++]; k]; Array[sn, 80] (* Harvey P. Dale, Aug 09 2023 *)
PROG
(Magma) S:=[]; for n in [1..72] do q:=n^4; k:=n+1; while not IsPrime(q+k^4) do k+:=1; end while; Append(~S, k); end for; S; // Klaus Brockhaus, Apr 12 2009
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 01 2009
EXTENSIONS
Edited and entries verified by Klaus Brockhaus, Apr 12 2009
Corrected by Harvey P. Dale, Aug 09 2023
STATUS
approved