

A126769


Primes p of the form k^4 + s where k > 1, s >= 1 and k^2 + s is also prime.


9



17, 19, 23, 29, 31, 41, 43, 53, 59, 71, 73, 79, 83, 89, 101, 103, 109, 113, 131, 139, 149, 151, 163, 173, 179, 181, 191, 193, 199, 211, 223, 229, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 337, 347, 349, 353, 359, 367, 379, 383, 389
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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

Cf. A128292.
Sequence in context: A054796 A322275 A008366 * A092216 A180948 A320869
Adjacent sequences: A126766 A126767 A126768 * A126770 A126771 A126772


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



