A228069 Signed pseudo characteristic function of primes by annihilation of composites up to p-1, here p=13 and sign (+).

1, 13, 29, 43, 53, 59, 71, 73, 79, 83, 89, 97, 101, 103, 149, 163, 169, 173, 179, 191, 193, 199, 211, 221, 223, 239, 263, 269, 281, 283, 289, 293, 299, 307, 311, 313, 331, 359, 373, 379, 383, 389, 401, 403, 409, 419, 421, 431, 433, 449, 479, 491, 493, 499, 503

Offset

1,2

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 (249 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 &] (* T. D. Noe, Aug 16 2013 *)

Crossrefs

Cf. A228070.

Sequence in context: A339113 A309356 A322551 * A351387 A044074 A044455

Adjacent sequences: A228066 A228067 A228068 * A228070 A228071 A228072

Keyword

nonn

Author

Freimut Marschner, Aug 08 2013

Status

approved