A247857 Primes of the form a^2 + b^4, with repetition.

2, 5, 17, 17, 37, 41, 97, 97, 101, 137, 181, 197, 241, 257, 257, 277, 281, 337, 337, 401, 457, 577, 617, 641, 641, 661, 677, 757, 769, 821, 857, 881, 881, 977, 1097, 1109, 1201, 1217, 1237, 1297, 1297, 1301, 1321, 1409, 1481, 1601, 1657, 1697, 1777, 2017, 2069, 2137, 2281, 2389, 2417, 2417, 2437

Offset: 1

Author

Jean-François Alcover, Sep 25 2014

Keywords

• nonn

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

Examples

			Since 97 = 4^2 + 3^4 = 9^2 + 2^4, it appears twice in the sequence.
		

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

Crossrefs

Cf. A002645, A028916 (same sequence without repetition).

Cf. A000040, A256852.

Programs

Haskell

a247857 n = a247857_list !! (n-1)
a247857_list = concat $ zipWith replicate a256852_list a000040_list
-- Reinhard Zumkeller, Apr 11 2015

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&)]