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 A228070 Signed pseudo characteristic function of primes by annihilation of composites up to p-1, here p=13 and sign (-). 2

%I

%S 17,19,23,31,37,41,47,61,67,107,109,113,127,131,137,139,151,157,167,

%T 181,197,227,229,233,241,247,251,257,271,277,317,323,337,347,349,353,

%U 361,367,377,391,397,437,439,443,457,461,463,467,481,487,527,533,547,557

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

%C 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.

%C 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.

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

%C 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.

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

%C 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.

%H Freimut Marschner and T. D. Noe, <a href="/A228070/b228070.txt">Table of n, a(n) for n = 1..1000</a> (205 terms from Freimut Marschner)

%F 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.

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

%Y Cf. A228069.

%K nonn

%O 1,1

%A _Freimut Marschner_, Aug 08 2013

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