A Sieve for Prime Based on Extension Form of Not Prime

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References

[1] Wikipedia.
http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

[2] A. O. L. Atkin and D. J. Bernstein, “Prime Sieves Using Binary Quadratic Forms,” Mathematics of Computation, Vol. 73, 2004, pp. 1023-1030.
doi:10.1090/S0025-5718-03-01501-1

[3] Wikipedia. http://en.wikipedia.org/wiki/Sieve_of_Atkin

[4] http://math.stackexchange.com/questions/141224/finite-sum-of-reciprocal-odd-integers

[5] T. M. Apostol, “Calculus I,” Bollati Boringhieri, 2003, p. 481.