A124598 Primes p of the form k^2+s where k > 1 and 1 <= s < (k+1)^2, such that q = k^4+s is prime and larger than p.

5, 7, 11, 17, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 101, 107, 109, 127, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 307, 311, 331, 337, 347

Offset

1,1

Comments

The terms of this sequence illustrate a special case of the conjecture from A126769.

Links

R. J. Cano, Table of n, a(n) for n = 1..10000

Example

5 = 2^2+1 is prime, 17 = 2^4+1 is a larger prime and 1 < 3^2, hence 5 is a term.
29 = 4^2+13 is prime, 269 = 4^4+13 is a larger prime and 13 < 5^2, hence 29 is a term.
805499 = 897^2+890 is prime, 647395643771 = 897^4+890 is a larger prime and 890 < 898^2, hence 805499 is a term.
Prime number 19 has the form k^2+s with s < (k+1)^2 in two ways, as 3^2+10 and 4^2+3. Neither 3^4+10 = 91 nor 4^4+3 = 259 is prime, hence 19 is not in the sequence.

Prog

(PARI) m=19; v=[]; for(k=2, m, for(s=1, (k+1)^2-1, if((p=k^2+s)<m^2&&isprime(p)&&(q=k^4+s)>p&&isprime(q), v=concat(v, p)))); print(Set(v)) \\
(PARI) upto(n)=my(res = List()); forprime(p = 5, n, for(k = ceil(sqrt(p / 2 + 1/4) - 0.5), sqrtint(p-1), if(isprime(k^4 + p - k^2), listput(res, p); next(2)))); res \\ David A. Corneth, Apr 08 2018

Crossrefs

Cf. A128292, A125283, A126769.

Sequence in context: A282739 A072249 A076665 * A096215 A144742 A059786

Adjacent sequences: A124595 A124596 A124597 * A124599 A124600 A124601

Keyword

nonn,easy

Author

Tomas Xordan, Mar 02 2007

Extensions

Edited, corrected and extended by Klaus Brockhaus, Mar 05 2007

Status

approved