Search: id:A103431
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%I A103431
%S A103431 1,1,2,3,2,3,1,4,2,5,1,6,4,5,7,2,7,5,6,3,8,5,8,4,9,1,10,3,10,7,8,11,4,
%T A103431 11,7,10,6,11,2,13,9,10,7,12,1,14,2,2,8,13,4,15,1,16,10,13,9,14,5,16,2,
%U A103431 17,12,13,11,14,9,16,5,18,8,17,19,7,18,10,17,6,19,1,20,3,20,14,15,12,17
%N A103431 Subsequence of the Gaussian primes, where only Gaussian primes a+bi with 
               a>0, b>=0 are listed. Ordered by the norm N(a+bi)=sqrt(a^2+b^2) and 
               the size of the real part, when the norms are equal. a(n) is the 
               real part of the Gaussian prime. Sequence A103432 gives the imaginary 
               parts.
%C A103431 Definition of Gaussian primes (Pieper, Die komplexen Zahlen, p. 122): 
               1) i+i, norm N(i+i) = sqrt(2) 2) Natural primes p with p = 3 mod 
               4, norm N(p) = p. 3) primes a+bi, a>0, b>0 with a^2 + b^2 = p = 1 
               mod 4, p natural prime. Norm N(a+bi) = sqrt(p). b+ai is a different 
               Gaussian prime number, b+ai can not be factored into a+bi and a unit. 
               4) All complex numbers from 1) to 3) multiplied by the units -1,i,
               -i, these are the associated numbers. The sequence contains all the 
               Gaussian primes mentioned in 1) - 3).
%C A103431 Every complex number can be factored completely into the Gaussian prime 
               numbers defined by the sequence, an additional unit as factor can 
               be necesarry. This factorization can be used to calculate the complex 
               sigma, as defined by Spira. The elements a(n) are ordered by the 
               size of their norm. If the two different primes a+bi and b+ai have 
               the same norm, they are ordered by the size of the real part of the 
               complex prime number. So a+bi follows b+ai in the sequence, if a 
               > b.
%C A103431 Of course this is not the only possible definition. As primes p = 1 mod 
               4 can be factored in p = (-i)(a+bi)(b+ai) and the norm N(a+bi) = 
               N(b+ai) = sqrt(p), these primes a+bi occur much earlier in the sequence 
               as p does in the sequence of natural primes. 4+5i with norm sqrt(37) 
               occurs before prime 7.
%D A103431 H. Pieper, "Die komplexen Zahlen", Verlag Harri Deutsch, p. 122
%D A103431 R. Spira, "The Complex Sum Of Divisors", American Mathematical Monthly, 
               1961 Vol. 68, p. 120-124
%Y A103431 Cf. A103432.
%Y A103431 Sequence in context: A004549 A026600 A106560 this_sequence A125928 A114388 
               A075789
%Y A103431 Adjacent sequences: A103428 A103429 A103430 this_sequence A103432 A103433 
               A103434
%K A103431 nonn
%O A103431 1,3
%A A103431 Sven Simon (sven-h.simon(AT)t-online.de), Feb 05 2005; corrected Feb 
               20 2005 and again on Aug 06 2006

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