A136618 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.

5, 89, 107, 139, 313, 331, 409, 421, 443, 449, 461, 491, 503, 547, 653, 757, 761, 769, 941, 947, 1063, 1181

Offset

1,1

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.

Links

Table of n, a(n) for n=1..22.

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]

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}]

Crossrefs

Sequence in context: A054954 A266064 A106971 * A138700 A139937 A059696

Adjacent sequences: A136615 A136616 A136617 * A136619 A136620 A136621

Keyword

nonn,uned

Author

Roger L. Bagula, Apr 14 2008

Status

approved