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%I #54 Aug 04 2021 03:15:01
%S 2,5,17,37,41,97,101,137,181,197,241,257,277,281,337,401,457,577,617,
%T 641,661,677,757,769,821,857,881,977,1097,1109,1201,1217,1237,1297,
%U 1301,1321,1409,1481,1601,1657,1697,1777,2017,2069,2137,2281,2389,2417,2437
%N Friedlander-Iwaniec primes: Primes of form a^2 + b^4.
%C John Friedlander and Henryk Iwaniec proved that there are infinitely many such primes.
%C A256852(A049084(a(n))) > 0. - _Reinhard Zumkeller_, Apr 11 2015
%C Primes in A111925. - _Robert Israel_, Oct 02 2015
%C Its intersection with A185086 is A262340, by the uniqueness part of Fermat's two-squares theorem. - _Jonathan Sondow_, Oct 05 2015
%C Cunningham calls these semi-quartan primes. - _Charles R Greathouse IV_, Aug 21 2017
%C Primes of the form (x^2 + y^2)/2, where x > y > 0, such that (x-y)/2 or (x+y)/2 is square. - _Thomas Ordowski_, Dec 04 2017
%C Named after the Canadian mathematician John Benjamin Friedlander (b. 1941) and the Polish-American mathematician Henryk Iwaniec (b. 1947). - _Amiram Eldar_, Jun 19 2021
%H T. D. Noe, <a href="/A028916/b028916.txt">Table of n, a(n) for n = 1..10000</a>
%H Art of Problem Solving, <a href="http://www.artofproblemsolving.com/wiki/index.php/Fermat's_Two_Squares_Theorem">Fermat's Two Squares Theorem</a>.
%H A. J. C. Cunningham, <a href="/wiki/File:High_quartan_factorisations_and_primes.pdf">High quartan factorisations and primes</a>, Messenger of Mathematics, Vol. 36 (1907), pp. 145-174.
%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>, Proc. Nat. Acad. Sci., Vol. 94 (1997), pp. 1054-1058.
%H J. Friedlander and H. Iwaniec, <a href="https://arxiv.org/abs/math/9811185">The polynomial x^2 + y^4 captures its primes</a>, arXiv:math/9811185 [math.NT], 1998; Ann. of Math. 148 (1998), 945-1040.
%H Charles R Greathouse IV, <a href="http://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes">Tables of special primes</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Friedlander%E2%80%93Iwaniec_theorem">Friedlander-Iwaniec theorem</a>.
%e 2 = 1^2 + 1^4.
%e 5 = 2^2 + 1^4.
%e 17 = 4^2 + 1^4 = 1^2 + 2^4.
%p N:= 10^5: # to get all terms <= N
%p S:= {seq(seq(a^2+b^4, a = 1 .. floor((N-b^4)^(1/2))),b=1..floor(N^(1/4)))}:
%p sort(convert(select(isprime,S),list)); # _Robert Israel_, Oct 02 2015
%t nn = 10000; t = {}; Do[n = a^2 + b^4; If[n <= nn && PrimeQ[n], AppendTo[t, n]], {a, Sqrt[nn]}, {b, nn^(1/4)}]; Union[t] (* _T. D. Noe_, Aug 06 2012 *)
%o (PARI) list(lim)=my(v=List([2]),t);for(a=1,sqrt(lim\=1),forstep(b=a%2+1, sqrtint(sqrtint(lim-a^2)), 2, t=a^2+b^4;if(isprime(t),listput(v,t)))); vecsort(Vec(v),,8) \\ _Charles R Greathouse IV_, Jun 12 2013
%o (Haskell)
%o a028916 n = a028916_list !! (n-1)
%o a028916_list = map a000040 $ filter ((> 0) . a256852) [1..]
%o -- _Reinhard Zumkeller_, Apr 11 2015
%Y Cf. A078523, A111925.
%Y Cf. A000290, A000583, A000040, A256852, A256863 (complement), A002645 (subsequence), subsequence of A247857.
%Y Primes of form n^2 + b^4, b fixed: A002496 (b = 1), A243451 (b = 2), A256775 (b = 3), A256776 (b = 4), A256777 (b = 5), A256834 (b = 6), A256835 (b = 7), A256836 (b = 8), A256837 (b = 9), A256838 (b = 10), A256839 (b = 11), A256840 (b = 12), A256841 (b = 13).
%K nonn
%O 1,1
%A _Warut Roonguthai_
%E Title expanded by _Jonathan Sondow_, Oct 02 2015