cp's OEIS Frontend

A266965 Primes of the form p = a^2 + b^2 where |a^2 - b^2| is composite.

%I A266965 #18 Jan 12 2016 00:11:47
%S A266965 17,29,37,41,53,73,89,97,101,109,113,137,149,157,173,193,197,229,233,
%T A266965 241,257,269,277,281,293,313,317,337,349,353,373,389,397,401,409,433,
%U A266965 449,457,461,509,521,541,557,569,577,593,601,613,617,641,653,661,673,677,701,709
%N A266965 Primes of the form p = a^2 + b^2 where |a^2 - b^2| is composite.
%C A266965 Inspired by A266954.
%C A266965 A266954 is a subsequence.
%C A266965 A067756 lists primes of the form p = a^2 + b^2 where |a^2 - b^2| is prime. So union of 2, A067756 and this sequence gives A002313. 2 is an exception because 1^2 - 1^2 = 0 is not prime or composite.
%e A266965 17 is a term because 4^2 + 1^2 = 17 is prime and 4^2 - 1^2 = 15 is composite.
%e A266965 29 is a term because 5^2 + 2^2 = 29 is prime and 5^2 - 2^2 = 21 is composite.
%e A266965 37 is a term because 6^2 + 1^2 = 37 is prime and 6^2 - 1^2 = 35 is composite.
%t A266965 lim = 50; Take[Select[Union@ Flatten@ Table[If[CompositeQ[Abs[a^2 - b^2]], a^2 + b^2, Nothing], {a, lim}, {b, lim}], PrimeQ], 56] (* _Michael De Vlieger_, Jan 07 2016 *)
%o A266965 (PARI) is(n) = {for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
%o A266965 f(p) = my(s=lift(sqrt(Mod(-1, p))), x=p, t); if(s>p/2, s=p-s); while(s^2>p, t=s; s=x%s; x=t); s;
%o A266965 forprime(p=3, 1e3, if(is(p) && !isprime(2*f(p)^2-p), print1(p, ", ")));
%o A266965 (PARI) list(lim) = my(v=List(), t); lim\=1; for(x=2, sqrtint(lim), for(y=1, min(sqrtint(lim-x^2), x), if(isprime(t=x^2+y^2) && !isprime(x^2-y^2), listput(v, t)))); vecsort(Vec(v), , 8)
%Y A266965 Cf. A002313, A067756, A266954.
%K A266965 nonn
%O A266965 1,1
%A A266965 _Altug Alkan_, Jan 07 2016