login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 1 07:45 EDT 2021. Contains 346384 sequences. (Running on oeis4.)