A228070 Signed pseudo characteristic function of primes by annihilation of composites up to p-1, here p=13 and sign (-).

17, 19, 23, 31, 37, 41, 47, 61, 67, 107, 109, 113, 127, 131, 137, 139, 151, 157, 167, 181, 197, 227, 229, 233, 241, 247, 251, 257, 271, 277, 317, 323, 337, 347, 349, 353, 361, 367, 377, 391, 397, 437, 439, 443, 457, 461, 463, 467, 481, 487, 527, 533, 547, 557

Offset

1,1

Comments

a(n) is defined by the sign of the product sin(n*Pi/2) *sin(n*Pi/3) *sin(n*Pi/5) *sin(n*Pi/7) *sin(n*Pi/11), where Pi is A000796.

This construction assigns values a(p)=0 to the primes up to p-1 (here p=13), values a(p)=-1 to the primes from p to p^2-1, and zeros to all composites up to p^2-1.

The offset has been set to p to eliminate the leading zeros.

The "pseudo" in the name indicates that this kind of Fourier synthesis (or sieve) starts to fail at n=169=p^2: a(169)=1 although 169 is a composite number.

The extrema of the sine function are prime numbers, while the zeros are the composite numbers annihilated in the interval [p,p^2[.

A generalization is to use the sign of sin(n*Pi/2) *sin(n*Pi/3)*... *sin(n*Pi/p) for an even higher number of sine factors, which works to indicate correctly primes and composites in the interval n=p to p^2-1.

Links

Freimut Marschner and T. D. Noe, Table of n, a(n) for n = 1..1000 (205 terms from Freimut Marschner)

Formula

Numbers n such that sign(sin(n*Pi/2) * sin(n*Pi/3) * sin(n*Pi/5) * sin(n*Pi/7) * sin(n*Pi/11)) = -1.

Mathematica

Select[Range[1000], Sign[Sin[#*Pi/2] * Sin[#*Pi/3] * Sin[#*Pi/5] * Sin[#*Pi/7] * Sin[#*Pi/11]] == -1 &]

Crossrefs

Cf. A228069.

Sequence in context: A007635 A140947 A205700 * A289685 A144487 A108266

Adjacent sequences: A228067 A228068 A228069 * A228071 A228072 A228073

Keyword

nonn

Author

Freimut Marschner, Aug 08 2013

Status

approved