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A349900 Primes of the form x^2 + (y^2+1)^2. 2

%I

%S 5,13,17,29,37,41,53,61,89,101,109,149,173,181,197,229,257,269,281,

%T 293,349,353,389,401,433,461,509,541,577,601,613,677,701,733,757,773,

%U 797,809,829,941,1049,1061,1093,1117,1181,1229,1297,1301

%N Primes of the form x^2 + (y^2+1)^2.

%C Merikoski proved that there are infinitely many primes of this form, and that the order of growth of the sequence up to x is x^(3/4)/log x. (His method did not provide enough Type II information to prove that there is a C such that there are ~ C*x^(3/4)/log x.)

%H Charles R Greathouse IV, <a href="/A349900/b349900.txt">Table of n, a(n) for n = 1..10000</a>

%H Jori Merikoski, <a href="https://arxiv.org/abs/2112.03617">The polynomials X^2+(Y^2+1)^2 and X^2+(Y^3+Z^3)^2 also capture their primes</a>, arXiv:2112.03617 [math.NT], 2021.

%F a(n) ≍ (n log n)^(4/3).

%o (PARI) list(lim)=my(v=List()); lim\=1; for(y=0,sqrtint(sqrtint(lim-1)-1), my(t=(y^2+1)^2); forstep(x=2-y%2,sqrtint(lim-t),2, my(p=x^2+t); if(isprime(p), listput(v,p)))); Set(v)

%o (PARI) is(n)=if(n<5 || !isprime(n), return(0)); for(y=0,sqrtint(sqrtint(n-1)-1), if(isprime(n-(y^2+1)^2), return(1))); 0

%Y Subsequence of A002144.

%Y Cf. A028916, A350687.

%K nonn

%O 1,1

%A _Charles R Greathouse IV_, Jan 11 2022

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Last modified August 17 23:13 EDT 2022. Contains 356204 sequences. (Running on oeis4.)