A319812 - OEIS

A319812

Square array read by antidiagonals: T(n,k) = (1 + i)-adic valuation of n + k*i, n >= 0, k >= 0, or -1 if n + k*i = 0.

%I #12 Oct 07 2018 08:35:06

%S -1,0,0,2,1,2,0,0,0,0,4,1,3,1,4,0,0,0,0,0,0,2,1,2,1,2,1,2,0,0,0,0,0,0,

%T 0,0,6,1,3,1,5,1,3,1,6,0,0,0,0,0,0,0,0,0,0,2,1,2,1,2,1,2,1,2,1,2,0,0,

%U 0,0,0,0,0,0,0,0,0,0,4,1,3,1,4,1,3,1,4,1,3,1,4

%N Square array read by antidiagonals: T(n,k) = (1 + i)-adic valuation of n + k*i, n >= 0, k >= 0, or -1 if n + k*i = 0.

%C Equivalently, T(n,k) = 2-adic valuation of n^2 + k^2.

%C Table is symmetric with respect to the main diagonal. For any Gaussian integer z = x + y*i, its (1 + i)-adic valuation is T(|x|,|y|) if z != 0 and +oo if z = 0.

%F For n + k*i != 0:

%F T(n,k) = v(n^2 + k^2, 2) where v(k, 2) = A007814(k) is the 2-adic valuation of k.

%F T(n,k) = 2*min{v(n, 2), v(k, 2)} if v(n, 2) != v(k, 2), otherwise v(n, 2) + v(k, 2) + 1. Here v(0, 2) = +oo.

%e Table begins

%e X 0 2 0 4 0 2 0 6 0 2 0 4 0 2 0 8 ...