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 A256852 Number of ways to write prime(n) = a^2 + b^4. 5
 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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS a(A049084(A028916(n))) > 0; a(A049084(A256863(n))) = 0; Conjecture: a(n) <= 2, empirically checked for the first 10^6 primes. 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 Friedlander and Iwaniec proved that a(n) > 0 infinitely often. - Jonathan Sondow, Oct 05 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. Wikipedia, Friedlander-Iwaniec theorem EXAMPLE First numbers n, such that a(n) > 0: .   k |  n |   prime(n)                    | a(n) . ----+----+-------------------------------+----- .   1 |  1 |    2 = 1^2 + 1^4              |   1 .   2 |  3 |    5 = 2^2 + 1^4              |   1 .   3 |  7 |   17 = 1^2 + 2^4 = 4^2 + 1^4  |   2 .   4 | 12 |   37 = 6^2 + 1^4              |   1 .   5 | 13 |   41 = 5^2 + 2^4              |   1 .   6 | 25 |   97 = 4^2 + 3^4 = 9^2 + 2^4  |   2 .   7 | 33 |  101 = 10^2 + 1^4             |   1 .   8 | 42 |  181 = 10^2 + 3^4             |   1 .   9 | 45 |  197 = 14^2 + 1^4             |   1 .  10 | 53 |  241 = 15^2 + 2^4             |   1 .  11 | 55 |  257 = 1^2 + 4^4 = 16^2 + 1^4 |   2 .  12 | 59 |  277 = 14^2 + 3^4             |   1 .  13 | 60 |  281 = 5^2 + 4^4              |   1 .  14 | 68 |  337 = 9^2 + 4^4 = 16^2 + 3^4 |   2 .  15 | 79 |  401 = 20^2 + 1^4             |   1 .  16 | 88 |  457 = 21^2 + 2^4             |   1 . PROG (Haskell) a256852 n = a256852_list !! (n-1) a256852_list = f a000040_list [] \$ tail a000583_list where    f ps'@(p:ps) us vs'@(v:vs)      | p > v     = f ps' (v:us) vs      | otherwise = (sum \$ map (a010052 . (p -)) us) : f ps us vs' CROSSREFS Cf. A000040, A000290, A000583, A010052, A002645, A028916, A256863. Sequence in context: A281154 A245536 A291203 * A128616 A270417 A266344 Adjacent sequences:  A256849 A256850 A256851 * A256853 A256854 A256855 KEYWORD nonn AUTHOR Reinhard Zumkeller, Apr 11 2015 STATUS approved

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