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A247857 Primes of the form a^2 + b^4, with repetition. 3
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 (list; graph; refs; listen; history; text; internal format)
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 * A025553 A075544 A104247

Adjacent sequences:  A247854 A247855 A247856 * A247858 A247859 A247860

KEYWORD

nonn

AUTHOR

Jean-Fran├žois Alcover, Sep 25 2014

STATUS

approved

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Last modified January 22 08:58 EST 2018. Contains 298042 sequences.