A124598 - OEIS

%I #35 Apr 18 2018 12:00:37

%S 5,7,11,17,23,29,31,37,41,43,47,53,59,61,67,71,79,83,89,97,101,107,

%T 109,127,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,

%U 227,229,233,239,241,251,257,263,269,271,277,281,283,307,311,331,337,347

%N 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.

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

%H R. J. Cano, <a href="/A124598/b124598.txt">Table of n, a(n) for n = 1..10000</a>

%e 5 = 2^2+1 is prime, 17 = 2^4+1 is a larger prime and 1 < 3^2, hence 5 is a term.

%e 29 = 4^2+13 is prime, 269 = 4^4+13 is a larger prime and 13 < 5^2, hence 29 is a term.

%e 805499 = 897^2+890 is prime, 647395643771 = 897^4+890 is a larger prime and 890 < 898^2, hence 805499 is a term.

%e 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.

%o (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)) \\

%o (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

%Y Cf. A128292, A125283, A126769.

%K nonn,easy

%O 1,1

%A _Tomas Xordan_, Mar 02 2007

%E Edited, corrected and extended by _Klaus Brockhaus_, Mar 05 2007