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%I #22 Jun 10 2018 14:51:01
%S 17,19,23,29,31,41,43,53,59,71,73,79,83,89,101,103,109,113,131,139,
%T 149,151,163,173,179,181,191,193,199,211,223,229,239,241,251,257,263,
%U 269,271,277,281,283,293,307,311,313,337,347,349,353,359,367,379,383,389
%N Primes p of the form k^4 + s where k > 1, s >= 1 and k^2 + s is also prime.
%C For primes not in this sequence see A128292.
%C Conjecture: Every prime q > 3 can be written in a nontrivial way as the sum of two or more squares, q = Sum_{i} (k_i)^2, such that the sum of the fourth powers of the squared numbers is again prime, p = Sum_{i} (k_i)^4. (Tomas Xordan)
%C This sequence illustrates an easy case of the conjecture: For primes q arising in the sequence there exists an integer k > 1, a positive integer s and a prime p such that k^2 < q, s = q - k^2, p = k^4 + s and p > q.
%C This corresponds to the case where only one of the squares is larger than 1 and all other s terms are equal to 1. - _M. F. Hasler_, May 23 2018
%H R. J. Cano, <a href="/A126769/b126769.txt">Table of n, a(n) for n = 1..10000</a>
%e 19 = 2^4+3 is prime and 2^2+3 = 7 is a smaller prime, hence 19 is a term.
%e 23 = 2^4+7 is prime and 2^2+7 = 11 is a smaller prime, hence 23 is a term.
%e 1307 = 6^4+11 is prime and 6^2+11 = 47 is a smaller prime, hence 1307 is a term.
%e 37 is prime, 2^4+21 is the only way to write 37 as k^4+s, but neither 2^2+21 = 25 nor 3^2+21 = 30 are prime, hence 37 is not in the sequence.
%o (PARI) {m=5;v=[];for(n=2,m,for(k=1,(m+1)^4,if(isprime(p=n^4+k)&&p<m^4&&(q=n^2+k)<p&&isprime(q),v=concat(v,p))));print(Set(v))} \\ _Klaus Brockhaus_, Feb 24 2007
%Y Cf. A128292.
%K nonn
%O 1,1
%A _Tomas Xordan_, Feb 16 2007
%E Edited, corrected and extended by _Klaus Brockhaus_, Feb 24 2007
%E Name edited following a suggestion from R. Sigrist, and the conjecture rephrased by _M. F. Hasler_, May 23 2018
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