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Search: id:A103431
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| A103431 |
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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. |
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+0
10
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| 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, 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, 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
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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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).
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.
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.
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REFERENCES
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H. Pieper, "Die komplexen Zahlen", Verlag Harri Deutsch, p. 122
R. Spira, "The Complex Sum Of Divisors", American Mathematical Monthly, 1961 Vol. 68, p. 120-124
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CROSSREFS
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Cf. A103432.
Sequence in context: A004549 A026600 A106560 this_sequence A125928 A114388 A075789
Adjacent sequences: A103428 A103429 A103430 this_sequence A103432 A103433 A103434
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KEYWORD
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nonn
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AUTHOR
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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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