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

1, 1, 2, 2, 3, 5, 7, 11, 17, 28, 43, 67, 108, 173, 272, 434, 690, 1115, 1772, 2815, 4528, 7267, 11646, 18799, 30378, 48956, 79270, 128267, 208509, 338533, 550262, 895284, 1457111, 2374753, 3874445, 6327042

Offset: 1

Author

Marek Wolf and Robert G. Wilson v, Jun 09 2013

Keywords

• nonn

Comments

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

Does not count double representations.

Links

• John Friedlander and Henryk Iwaniec, Using a parity-sensitive sieve to count prime values of a polynomial, PNAS February 18, 1997 94 (4) 1054-1058.

Crossrefs

Cf. A028916, A226495, A226496, A226497.

Programs

Mathematica

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}]

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