A stable set of primes of the form j^4 + k^4 with all j and k distinct.

97, 881, 1297, 12401, 49297, 54721, 119617, 166561, 260017, 317777, 462097, 961937, 988417, 1089841, 1800577, 2070241, 3959297, 5978801, 6015697, 6790897, 7813777, 8812241, 9075761, 11002177, 13382737, 14304097, 15417121, 17089297, 18724801, 23926961, 27685121

1

The sequence S000618 lists the pairs (j,k) for each number here. This sequence is discussed briefly in the Greenfield and Greenfield paper. It appears that higher powers, such as 8, do not work with the greedy algorithm. For the power 8, the greedy algorithm stops on the number 78 with 29 addition numbers unused. Higher powers appear to have similar problems.

T. D. Noe, Plot of 1000 terms

T. D. Noe, Table of 1000 terms

L. E. Greenfield and S. J. Greenfield, Some Problems of Combinatorial Number Theory Related to Bertrand's Postulate, J. Integer Sequences, 1998, #98.1.2.

(Mma) n = 200; lst = Reverse[Range[2 n]]; prms = {}; Do[m = lst[[1]]; lst = Delete[lst, 1]; pos = 1; While[p = m^4 + lst[[pos]]^4; Not[PrimeQ[p]], pos++]; AppendTo[prms, p]; lst = Delete[lst, pos], {i, n}]; Take[Sort[prms], n/2]

Cf. A091652, S000615, S000618.

nonn

T. D. Noe, May 05 2015