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

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, 16, 13, 14, 16, 17, 13, 14, 16, 18, 17, 19, 18, 17, 19, 20, 20, 15, 17, 20, 21, 19, 22, 20, 23, 21, 19, 20, 24, 23, 24, 18, 19, 25, 22, 25, 23, 26, 26, 22, 27, 26, 20

Offset: 1

Author

M. F. Hasler, Mar 09 2018

Keywords

• nonn
• look

Comments

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).

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.

Links

• M. F. Hasler, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

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.

Formula

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

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