1,1

For primes not in this sequence see A128292.

Conjecture: (primality conservation) Every prime q > 3 can in at least one nontrivial way be written as the sum of two or more squares, such that the sum p of the fourth powers of the squared numbers is also prime. q = sum{i}(a_i)^2, p = sum{i}(a_i)^4, p > q. (Tomas Xordan)

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.

19 = 2^4+3 is prime and 2^2+3 = 7 is a smaller prime, hence 19 is a term.

23 = 2^4+7 is prime and 2^2+7 = 11 is a smaller prime, hence 23 is a term.

1307 = 6^4+11 is prime and 6^2+11 = 47 is a smaller prime, hence 1307 is a term.

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.

(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)))); v=listsort(List(v), 1); for(j=1, #v, if(v[j]<=m^4, print1(v[j], ", ")))} /* Klaus Brockhaus, Feb 24 2007 */

Cf. A128292.

Sequence in context: A139049 A054796 A008366 this_sequence A092216 A106933 A106932

Adjacent sequences: A126766 A126767 A126768 this_sequence A126770 A126771 A126772

nonn

Tomas Xordan (xordan.tom(AT)gmail.com), Feb 16 2007

Edited, corrected and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Feb 24 2007