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

1,1

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

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

Cf. A000040, A256852.

Sequence in context: A057282 A276767 A123364 * A367784 A025553 A343775

Adjacent sequences: A247854 A247855 A247856 * A247858 A247859 A247860

Keyword

nonn

Author

Jean-François Alcover, Sep 25 2014

Status

approved