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A300587 Real part of the n-th Gaussian prime x + i*y, x >= y >= 0, ordered by norm x^2 + y^2. 4


%S 1,2,3,3,4,5,6,5,7,7,6,8,8,9,10,10,8,11,11,10,11,13,10,12,14,15,13,15,

%T 16,13,14,16,17,13,14,16,18,17,19,18,17,19,20,20,15,17,20,21,19,22,20,

%U 23,21,19,20,24,23,24,18,19,25,22,25,23,26,26,22,27,26,20

%N Real part of the n-th Gaussian prime x + i*y, x >= y >= 0, ordered by norm x^2 + y^2.

%C With the restriction Re(z) >= Im(z) >= 0 used here and in A239621, there is exactly one Gaussian prime z for each possible norm |z|^2 in A055025. Sequence A239397 lists both, (x, y) and (y, x), for each of these having x > y (i.e., except for x = y = 1).

%C The nice graph shows that the values are denser towards the upper bound a(n) <= sqrt(A055025(n)) ~ sqrt(2n log n) than to the lower bound sqrt(A055025(n)/2) ~ sqrt(n log n), while for the imaginary parts A300588, i.e., min(Re(z),Im(z)), the distribution looks rather uniform.

%H M. F. Hasler, <a href="/A300587/b300587.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A239621(2n-1) = A239397(4n-2) (= A239397(4n-5) for n > 1).

%F a(n) = sqrt(A055025(n) - A300588(n)^2).

%o (PARI) c=1; for(n=1,oo, matsize(f=factor(n*I))[1]<=2 && vecsum(f[,2])==2+(f[1, 1]==I) && !write("/tmp/b300587.txt",c" "max(real(f=f[3-f[1,2],1]),imag(f))) && c++>1e4 && break) \\ Replace write("/tmp/b300587.txt",c" by print1(", to print the values.

%Y Odd bisection of A239621. See A300588 for imaginary parts, A055025 for the norms.

%K nonn,look

%O 1,2

%A _M. F. Hasler_, Mar 09 2018

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Last modified September 18 04:22 EDT 2021. Contains 347508 sequences. (Running on oeis4.)