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A028916 Friedlander-Iwaniec primes: Primes of form a^2 + b^4. 32
2, 5, 17, 37, 41, 97, 101, 137, 181, 197, 241, 257, 277, 281, 337, 401, 457, 577, 617, 641, 661, 677, 757, 769, 821, 857, 881, 977, 1097, 1109, 1201, 1217, 1237, 1297, 1301, 1321, 1409, 1481, 1601, 1657, 1697, 1777, 2017, 2069, 2137, 2281, 2389, 2417, 2437 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

John Friedlander and Henryk Iwaniec proved that there are infinitely many such primes.

A256852(A049084(a(n))) > 0. - Reinhard Zumkeller, Apr 11 2015

Primes in A111925. - Robert Israel, Oct 02 2015

Its intersection with A185086 is A262340, by the uniqueness part of Fermat's two-squares theorem. - Jonathan Sondow, Oct 05 2015

Cunningham calls these semi-quartan primes. - Charles R Greathouse IV, Aug 21 2017

Primes of the form (x^2 + y^2)/2, where x > y > 0, such that (x-y)/2 or (x+y)/2 is square. - Thomas Ordowski, Dec 04 2017

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Art of Problem Solving, Fermat's Two Squares Theorem

A. J. C. Cunningham, High quartan factorisations and primes, Messenger of Mathematics 36 (1907), pp. 145-174.

John Friedlander and Henryk Iwaniec, Using a parity-sensitive sieve to count prime values of a polynomial, Proc. Nat. Acad. Sci. 94  (1997), 1054-1058.

Charles R Greathouse IV, Tables of special primes

Wikipedia, Friedlander-Iwaniec theorem

EXAMPLE

2 = 1^2 + 1^4.

5 = 2^2 + 1^4.

17 = 4^2 + 1^4 = 1^2 + 2^4.

MAPLE

N:= 10^5: # to get all terms <= N

S:= {seq(seq(a^2+b^4, a = 1 .. floor((N-b^4)^(1/2))), b=1..floor(N^(1/4)))}:

sort(convert(select(isprime, S), list)); # Robert Israel, Oct 02 2015

MATHEMATICA

nn = 10000; t = {}; Do[n = a^2 + b^4; If[n <= nn && PrimeQ[n], AppendTo[t, n]], {a, Sqrt[nn]}, {b, nn^(1/4)}]; Union[t] (* T. D. Noe, Aug 06 2012 *)

PROG

(PARI) list(lim)=my(v=List([2]), t); for(a=1, sqrt(lim\=1), forstep(b=a%2+1, sqrtint(sqrtint(lim-a^2)), 2, t=a^2+b^4; if(isprime(t), listput(v, t)))); vecsort(Vec(v), , 8) \\ Charles R Greathouse IV, Jun 12 2013

(Haskell)

a028916 n = a028916_list !! (n-1)

a028916_list = map a000040 $ filter ((> 0) . a256852) [1..]

-- Reinhard Zumkeller, Apr 11 2015

CROSSREFS

Cf. A078523, A111925.

Cf. A000290,  A000583, A000040, A256852, A256863 (complement), A002645 (subsequence), subsequence of A247857.

Primes of form n^2 + b^4, b fixed: A002496 (b = 1), A243451 (b = 2), A256775 (b = 3), A256776 (b = 4), A256777 (b = 5), A256834 (b = 6), A256835 (b = 7), A256836 (b = 8), A256837 (b = 9), A256838 (b = 10), A256839 (b = 11), A256840 (b = 12), A256841 (b = 13).

Sequence in context: A025537 A245784 A247068 * A100272 A107630 A078523

Adjacent sequences:  A028913 A028914 A028915 * A028917 A028918 A028919

KEYWORD

nonn

AUTHOR

Warut Roonguthai

EXTENSIONS

Title expanded by Jonathan Sondow, Oct 02 2015

STATUS

approved

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Last modified September 23 02:13 EDT 2018. Contains 315271 sequences. (Running on oeis4.)