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)=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.

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, 15, 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

Offset: 1

Author

Sven Simon, Feb 05 2005; corrected Feb 20 2005 and again on Aug 06 2006

Keywords

• nonn

Comments

Definition of Gaussian primes (Pieper, Die komplexen Zahlen, p. 122): 1) i+i, norm N(i+i) = 2. 2) Natural primes p with p = 3 mod 4, norm N(p) = p^2. 3) primes a+bi, a>0, b>0 with a^2 + b^2 = p = 1 mod 4, p natural prime. Norm N(a+bi) = p. b+ai is a different Gaussian prime number, b+ai cannot 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 necessary. 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) = p, these primes a+bi occur much earlier in the sequence than p does in the sequence of natural primes. 4+5i with norm 41 occurs before prime 7 with norm 49.

References

• H. Pieper, "Die komplexen Zahlen", Verlag Harri Deutsch, p. 122

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Sven Simon, List with Gaussian primes (extended) of A103431/A103432
• R. Spira, The Complex Sum Of Divisors, American Mathematical Monthly, 1961 Vol. 68, pp. 120-124.
• Wikipedia, Table of Gaussian integer factorizations

Crossrefs

Cf. A103432, A055025.

Programs

Maple

N:= 100: # to get all terms with norm <= N
p1:= select(isprime,[seq(i,i=3..N,4)]):
p2:= select(isprime,[seq(i,i=1..N^2,4)]):
p2:= map(t -> GaussInt:-GIfactors(t)[2][1][1],p2):
p3:= sort( [1+I, op(p1),op(p2)],(a,b) -> Re(a)^2 + Im(a)^2  < Re(b)^2 + Im(b)^2):
g:= proc(z)
    local a,b;
    a:= Re(z); b:= Im(z);
    if b = 0 then z
    else
      a:= abs(a);
      b:= abs(b);
      if a = b then a
      elif a < b then a,b
      else b,a
      fi
    fi
end proc:
map(g, p3); # Robert Israel, Feb 23 2016

Mathematica

maxNorm = 500;
norm[z_] := Re[z]^2 + Im[z]^2;
m = Sqrt[maxNorm] // Ceiling;
gp = Select[Table[a + b I, {a, 1, m}, {b, 0, m}] // Flatten, norm[#] <= maxNorm && PrimeQ[#, GaussianIntegers -> True]&];
SortBy[gp, norm[#] maxNorm + Abs[Re[#]]&] // Re (* Jean-François Alcover, Mar 04 2019 *)

Extensions

Edited (mostly to correct meaning of norm) by Franklin T. Adams-Watters, Mar 04 2011

a(48) corrected by Robert Israel, Feb 23 2016