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%I #14 Oct 05 2015 10:12:13
%S 1,0,1,0,0,0,2,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,
%T 0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,2,0,0,0,1,1,0,0,0,0,0,0,0,2,
%U 0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0
%N Number of ways to write prime(n) = a^2 + b^4.
%C a(A049084(A028916(n))) > 0; a(A049084(A256863(n))) = 0;
%C Conjecture: a(n) <= 2, empirically checked for the first 10^6 primes.
%C The conjecture is true, because by the uniqueness part of Fermat's two-squares theorem, at most one duplicate of a^2 + b^4 can exist. Namely, if a is a square, say a = B^2, then a^2 + b^4 = A^2 + B^4 where A = b^2. - _Jonathan Sondow_, Oct 03 2015
%C Friedlander and Iwaniec proved that a(n) > 0 infinitely often. - _Jonathan Sondow_, Oct 05 2015
%H Reinhard Zumkeller, <a href="/A256852/b256852.txt">Table of n, a(n) for n = 1..10000</a>
%H Art of Problem Solving, <a href="http://www.artofproblemsolving.com/wiki/index.php/Fermat's_Two_Squares_Theorem">Fermat's Two Squares Theorem</a>
%H John Friedlander and Henryk Iwaniec, <a href="http://www.pnas.org/cgi/content/full/94/4/1054">Using a parity-sensitive sieve to count prime values of a polynomial</a>, PNAS, vol. 94 no. 4, pp. 1054-1058.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Friedlander%E2%80%93Iwaniec_theorem">Friedlander-Iwaniec theorem</a>
%e First numbers n, such that a(n) > 0:
%e . k | n | prime(n) | a(n)
%e . ----+----+-------------------------------+-----
%e . 1 | 1 | 2 = 1^2 + 1^4 | 1
%e . 2 | 3 | 5 = 2^2 + 1^4 | 1
%e . 3 | 7 | 17 = 1^2 + 2^4 = 4^2 + 1^4 | 2
%e . 4 | 12 | 37 = 6^2 + 1^4 | 1
%e . 5 | 13 | 41 = 5^2 + 2^4 | 1
%e . 6 | 25 | 97 = 4^2 + 3^4 = 9^2 + 2^4 | 2
%e . 7 | 33 | 101 = 10^2 + 1^4 | 1
%e . 8 | 42 | 181 = 10^2 + 3^4 | 1
%e . 9 | 45 | 197 = 14^2 + 1^4 | 1
%e . 10 | 53 | 241 = 15^2 + 2^4 | 1
%e . 11 | 55 | 257 = 1^2 + 4^4 = 16^2 + 1^4 | 2
%e . 12 | 59 | 277 = 14^2 + 3^4 | 1
%e . 13 | 60 | 281 = 5^2 + 4^4 | 1
%e . 14 | 68 | 337 = 9^2 + 4^4 = 16^2 + 3^4 | 2
%e . 15 | 79 | 401 = 20^2 + 1^4 | 1
%e . 16 | 88 | 457 = 21^2 + 2^4 | 1 .
%o (Haskell)
%o a256852 n = a256852_list !! (n-1)
%o a256852_list = f a000040_list [] $ tail a000583_list where
%o f ps'@(p:ps) us vs'@(v:vs)
%o | p > v = f ps' (v:us) vs
%o | otherwise = (sum $ map (a010052 . (p -)) us) : f ps us vs'
%Y Cf. A000040, A000290, A000583, A010052, A002645, A028916, A256863.
%K nonn
%O 1,7
%A _Reinhard Zumkeller_, Apr 11 2015
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