A300588 Imaginary part y of the n-th Gaussian prime x + i*y, x >= y >= 0, ordered by norm x^2 + y^2 = A055025(n)^2.

1, 1, 0, 2, 1, 2, 1, 4, 0, 2, 5, 3, 5, 4, 1, 3, 7, 0, 4, 7, 6, 2, 9, 7, 1, 2, 8, 4, 1, 10, 9, 5, 2, 12, 11, 9, 5, 8, 0, 7, 10, 6, 1, 3, 14, 12, 7, 4, 10, 5, 11, 0, 10, 14, 13, 1, 8, 5, 17, 16, 4, 13, 6, 12, 1, 5, 15, 2, 9, 19, 12, 17, 11, 5, 14, 10, 18, 4, 6, 16, 20, 19, 10, 13, 0, 4, 6

Offset: 1

Author

M. F. Hasler, Mar 09 2018

Keywords

• nonn

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).

According to the graph, the values seem rather uniformly distributed between 0 and the upper bound sqrt(A055025(n)/2) ~ sqrt(n log n), in contrast to the values of the real parts A300587(n).

Examples

			The smallest Gaussian primes with Re(z) >= Im(z) >= 0, ordered by norm, are 1+i, 2+i, 3, 3+i, ...
Their imaginary parts, listed here, are a(1) = 1, a(2) = 1, a(3) = 0, a(4) = 1,
		

Links

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

Crossrefs

Even bisection of A239621. See A300587 for real 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/b300588.txt",c" "min(real(f=f[3-f[1,2],1]),imag(f))) && c++>1e4 && break) \\ Replace write("/tmp/b300588.txt",c" by print1(" to print the values.

Formula

a(n) = A239621(2n) = A239397(4n-3) (= A239397(4n-4) for n > 1).

a(n) = sqrt(A055025(n) - A300587(n)^2).