This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A256852 #14 Oct 05 2015 10:12:13 %S A256852 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 A256852 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 A256852 0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0 %N A256852 Number of ways to write prime(n) = a^2 + b^4. %C A256852 a(A049084(A028916(n))) > 0; a(A049084(A256863(n))) = 0; %C A256852 Conjecture: a(n) <= 2, empirically checked for the first 10^6 primes. %C A256852 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 A256852 Friedlander and Iwaniec proved that a(n) > 0 infinitely often. - _Jonathan Sondow_, Oct 05 2015 %H A256852 Reinhard Zumkeller, <a href="/A256852/b256852.txt">Table of n, a(n) for n = 1..10000</a> %H A256852 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 A256852 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 A256852 Wikipedia, <a href="http://en.wikipedia.org/wiki/Friedlander%E2%80%93Iwaniec_theorem">Friedlander-Iwaniec theorem</a> %e A256852 First numbers n, such that a(n) > 0: %e A256852 . k | n | prime(n) | a(n) %e A256852 . ----+----+-------------------------------+----- %e A256852 . 1 | 1 | 2 = 1^2 + 1^4 | 1 %e A256852 . 2 | 3 | 5 = 2^2 + 1^4 | 1 %e A256852 . 3 | 7 | 17 = 1^2 + 2^4 = 4^2 + 1^4 | 2 %e A256852 . 4 | 12 | 37 = 6^2 + 1^4 | 1 %e A256852 . 5 | 13 | 41 = 5^2 + 2^4 | 1 %e A256852 . 6 | 25 | 97 = 4^2 + 3^4 = 9^2 + 2^4 | 2 %e A256852 . 7 | 33 | 101 = 10^2 + 1^4 | 1 %e A256852 . 8 | 42 | 181 = 10^2 + 3^4 | 1 %e A256852 . 9 | 45 | 197 = 14^2 + 1^4 | 1 %e A256852 . 10 | 53 | 241 = 15^2 + 2^4 | 1 %e A256852 . 11 | 55 | 257 = 1^2 + 4^4 = 16^2 + 1^4 | 2 %e A256852 . 12 | 59 | 277 = 14^2 + 3^4 | 1 %e A256852 . 13 | 60 | 281 = 5^2 + 4^4 | 1 %e A256852 . 14 | 68 | 337 = 9^2 + 4^4 = 16^2 + 3^4 | 2 %e A256852 . 15 | 79 | 401 = 20^2 + 1^4 | 1 %e A256852 . 16 | 88 | 457 = 21^2 + 2^4 | 1 . %o A256852 (Haskell) %o A256852 a256852 n = a256852_list !! (n-1) %o A256852 a256852_list = f a000040_list [] $ tail a000583_list where %o A256852 f ps'@(p:ps) us vs'@(v:vs) %o A256852 | p > v = f ps' (v:us) vs %o A256852 | otherwise = (sum $ map (a010052 . (p -)) us) : f ps us vs' %Y A256852 Cf. A000040, A000290, A000583, A010052, A002645, A028916, A256863. %K A256852 nonn %O A256852 1,7 %A A256852 _Reinhard Zumkeller_, Apr 11 2015