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.

-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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 1, 3, 1, 4, 1, 3, 1, 4, 1, 3, 1, 4

Offset

0,4

Comments

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

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.

Links

Table of n, a(n) for n=0..90.

Formula

For n + k*i != 0:

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

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.

Example

Table begins
X 0 2 0 4 0 2 0 6 0 2 0 4 0 2 0 8 ...
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ...
2 0 3 0 2 0 3 0 2 0 3 0 2 0 3 0 2 ...
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ...
4 0 2 0 5 0 2 0 4 0 2 0 5 0 2 0 4 ...
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ...
2 0 3 0 2 0 3 0 2 0 3 0 2 0 3 0 2 ...
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ...
6 0 2 0 4 0 2 0 7 0 2 0 4 0 2 0 6 ...
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ...
2 0 3 0 2 0 3 0 2 0 3 0 2 0 3 0 2 ...
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ...
4 0 2 0 5 0 2 0 4 0 2 0 5 0 2 0 4 ...
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ...
2 0 3 0 2 0 3 0 2 0 3 0 2 0 3 0 2 ...
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ...
8 0 2 0 4 0 2 0 6 0 2 0 4 0 2 0 9 ...
...
(X denotes that (1 + i)-adic valuation of 0 is +oo)

Prog

(PARI) T(n, k) = valuation(n^2+k^2, 2)

Crossrefs

Cf. A007814.

Sequence in context: A070965 A079548 A175620 * A079071 A322795 A050602

Adjacent sequences: A319809 A319810 A319811 * A319813 A319814 A319815

Keyword

sign,easy,tabl

Author

Jianing Song, Sep 28 2018

Status

approved