OFFSET
1,1
COMMENTS
For primes not in this sequence see A128292.
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)
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.
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
LINKS
R. J. Cano, Table of n, a(n) for n = 1..10000
EXAMPLE
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.
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Tomas Xordan, Feb 16 2007
EXTENSIONS
Edited, corrected and extended by Klaus Brockhaus, Feb 24 2007
Name edited following a suggestion from R. Sigrist, and the conjecture rephrased by M. F. Hasler, May 23 2018
STATUS
approved