OFFSET
1,1
COMMENTS
Duplicates, which begin 17, 97, 257, 337, etc, are quartan primes A002645, except 2 (noticed by Michel Marcus).
Is there any triple?
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
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Art of Problem Solving, Fermat's Two Squares Theorem
John Friedlander and Henryk Iwaniec, Using a parity-sensitive sieve to count prime values of a polynomial, PNAS, vol. 94 no. 4, pp. 1054-1058.
Marek Wolf, Continued fractions constructed from prime numbers, arXiv:1003.4015 [math.NT], 2010, p. 8.
Wikipedia, Friedlander-Iwaniec theorem
EXAMPLE
Since 97 = 4^2 + 3^4 = 9^2 + 2^4, it appears twice in the sequence.
MATHEMATICA
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&)]
PROG
(Haskell)
a247857 n = a247857_list !! (n-1)
a247857_list = concat $ zipWith replicate a256852_list a000040_list
-- Reinhard Zumkeller, Apr 11 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Jean-François Alcover, Sep 25 2014
STATUS
approved