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%I #3 Mar 30 2012 17:34:23
%S 5,89,107,139,313,331,409,421,443,449,461,491,503,547,653,757,761,769,
%T 941,947,1063,1181
%N Primes that give answers to the find root solution the equation of that are imaginary part less than zero: x^2 + 2 x^(3/2 - 4*I*Pi x) + x^(1 - 8*I*Pi* x) == 0.
%C Derivation is like this:
%C z = 1/2 + I*4*Pi*x
%C y = ExpandAll[x^2*((1 + x^(-z)))/(1 - x^(-z))*((1 + x^(-z)))]
%C y has upper part of:
%C x^2 + 2 x^(3/2 - 4*I*Pi x) + x^(1 - 8*I*Pi* x)
%C The find root the limiting zeros of this equation near as
%C Prime[n] starting points. The Im[x]<0 results gives two specific solutions
%C that are unlike the others. Most of the first type of solutions are on a specific curve.
%C The two Im[x]<0 solutions are specifically:
%C {{0.275165+I*( -0.517457)}, {0.701928+I*( -0.0217616)}}
%C x=Prime[n]+Delta1+I*Delta2: Delta2 small and approaching a limit as n->Large
%C In total there are three types of solutions.
%F a(n)=If x such that FindRoot[x^2 + 2 x^(3/2 - 4*I*Pi x) + x^(1 - 8*I*Pi* x) == 0, {x, Prime[n]}] has Imaginary part Im[x]<0, report the prime[n]
%t a1 = Flatten[Table[If[(Im[x] /. FindRoot[x^2 + 2 x^(3/2 - 4*I*Pi x) + x^(1 - 8*I*Pi* x) == 0, {x, Prime[n]}]) < 0, Prime[n], {}], {n, 1, 200}]
%K nonn,uned
%O 1,1
%A _Roger L. Bagula_, Apr 14 2008
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