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A226498 The number of primes of the form i^2 + j^4 (A028916) <= 2^n. 3

%I #24 Dec 11 2020 11:31:31

%S 1,1,2,2,3,5,7,11,17,28,43,67,108,173,272,434,690,1115,1772,2815,4528,

%T 7267,11646,18799,30378,48956,79270,128267,208509,338533,550262,

%U 895284,1457111,2374753,3874445,6327042

%N The number of primes of the form i^2 + j^4 (A028916) <= 2^n.

%C Iwaniec and Friedlander proved there are infinity of the primes of the form i^2+j^4, and hence a(n) increases without bound.

%C Does not count double representations.

%H John Friedlander and Henryk Iwaniec, <a href="https://doi.org/10.1073/pnas.94.4.1054">Using a parity-sensitive sieve to count prime values of a polynomial</a>, PNAS February 18, 1997 94 (4) 1054-1058.

%t mx = 2^40; lst = {}; Do[a = i^2 + j^4; If[ PrimeQ[a], AppendTo[lst, a]], {i, Sqrt[mx]}, {j, Sqrt[ Sqrt[mx - i^2]]}]; Table[ Length@ Select[ Union@ lst, # < 2^n &], {n, 40}]

%o (PARI) a(n)=my(N=2^n,v=List(),t);for(a=1,sqrt(N),forstep(b=a%2+1, sqrtint(sqrtint(N-a^2)), 2, t=a^2+b^4;if(isprime(t),listput(v,t)))); 1+#vecsort(Vec(v),,8) \\ _Charles R Greathouse IV_, Jun 12 2013

%Y Cf. A028916, A226495, A226496, A226497.

%K nonn

%O 1,3

%A Marek Wolf and _Robert G. Wilson v_, Jun 09 2013

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)