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%I #29 Oct 05 2015 09:19:36
%S 2,5,17,17,37,41,97,97,101,137,181,197,241,257,257,277,281,337,337,
%T 401,457,577,617,641,641,661,677,757,769,821,857,881,881,977,1097,
%U 1109,1201,1217,1237,1297,1297,1301,1321,1409,1481,1601,1657,1697,1777,2017,2069,2137,2281,2389,2417,2417,2437
%N Primes of the form a^2 + b^4, with repetition.
%C Duplicates, which begin 17, 97, 257, 337, etc, are quartan primes A002645, except 2 (noticed by _Michel Marcus_).
%C Is there any triple?
%C No, by the uniqueness part of Fermat's two-squares theorem, at most one duplicate of a^2 + b^4 can exist. Namely, when a is a square, say a = B^2, then a^2 + b^4 = A^2 + B^4 where A = b^2. (This also proves Marcus's comment, since a^2 + b^4 = b^4 + B^4.) - _Jonathan Sondow_, Oct 03 2015
%H Reinhard Zumkeller, <a href="/A247857/b247857.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 John Friedlander and Henryk Iwaniec, <a href="http://www.pnas.org/cgi/content/full/94/4/1054">Using a parity-sensitive sieve to count prime values of a polynomial</a>, PNAS, vol. 94 no. 4, pp. 1054-1058.
%H Marek Wolf, <a href="http://arxiv.org/abs/1003.4015">Continued fractions constructed from prime numbers</a>, arXiv:1003.4015 [math.NT], 2010, p. 8.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Friedlander%E2%80%93Iwaniec_theorem">Friedlander-Iwaniec theorem</a>
%e Since 97 = 4^2 + 3^4 = 9^2 + 2^4, it appears twice in the sequence.
%t max = 10^4; r = Reap[Do[n = a^2 + b^4; If[n <= max && PrimeQ[n], Sow[n]], {a, Sqrt[max]}, {b, max^(1/4)}]][[2, 1]]; Union[r, SameTest -> (False&)]
%o (Haskell)
%o a247857 n = a247857_list !! (n-1)
%o a247857_list = concat $ zipWith replicate a256852_list a000040_list
%o -- _Reinhard Zumkeller_, Apr 11 2015
%Y Cf. A002645, A028916 (same sequence without repetition).
%Y Cf. A000040, A256852.
%K nonn
%O 1,1
%A _Jean-François Alcover_, Sep 25 2014
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