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A136618
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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.
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0
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5, 89, 107, 139, 313, 331, 409, 421, 443, 449, 461, 491, 503, 547, 653, 757, 761, 769, 941, 947, 1063, 1181
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Derivation is like this:
z = 1/2 + I*4*Pi*x
y = ExpandAll[x^2*((1 + x^(-z)))/(1 - x^(-z))*((1 + x^(-z)))]
y has upper part of:
x^2 + 2 x^(3/2 - 4*I*Pi x) + x^(1 - 8*I*Pi* x)
The find root the limiting zeros of this equation near as
Prime[n] starting points. The Im[x]<0 results gives two specific solutions
that are unlike the others. Most of the first type of solutions are on a specific curve.
The two Im[x]<0 solutions are specifically:
{{0.275165+I*( -0.517457)}, {0.701928+I*( -0.0217616)}}
x=Prime[n]+Delta1+I*Delta2: Delta2 small and approaching a limit as n->Large
In total there are three types of solutions.
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FORMULA
| 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]
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MATHEMATICA
| 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}]
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CROSSREFS
| Sequence in context: A184239 A054954 A106971 * A138700 A139937 A059696
Adjacent sequences: A136615 A136616 A136617 * A136619 A136620 A136621
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KEYWORD
| nonn,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 14 2008
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