A323377 Square array read by ascending antidiagonals: T(n,k) = Kronecker(prime(n)/prime(k)), n, k >= 1.

0, -1, -1, -1, 0, -1, 1, -1, -1, 1, -1, 1, 0, -1, -1, -1, -1, -1, -1, 1, -1, 1, 1, 1, 0, 1, 1, 1, -1, -1, -1, 1, -1, -1, -1, -1, 1, 1, -1, -1, 0, -1, -1, -1, 1, -1, -1, 1, -1, -1, -1, -1, 1, 1, -1, 1, -1, -1, -1, -1, 0, -1, 1, -1, -1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, -1, -1

Offset: 1

Author

Jianing Song, Jan 12 2019

Keywords

• sign
• tabl

Comments

The n-th row is the same as the n-th column if and only if n = 1 or prime(n) == 1 (mod 4).

In general, for any m != 0 and n > 0, Kronecker symbol (m/n) can be written as the product of the terms of this table and the terms of the form (-1/p) where p is any prime.

According to Chebyshev's bias, there seem to be more -1's than 1's among the first terms of any row or any column. One can see from the table in the example section that there are 54 -1's and 36 1's in the upper left 10 X 10 square of the table. There are 5158 -1's and 4742 1's in the upper left 100 X 100 square of the table.

Examples

			Table begins
     |  k  |  1   2   3   4   5   6   7   8   9  10  ...
   n | p() |  2   3   5   7  11  13  17  19  23  29  ...
  ---+-----+--------------------------------------------
   1 |   2 |  0, -1, -1,  1, -1, -1,  1, -1,  1, -1, ...
   2 |   3 | -1,  0, -1, -1,  1,  1, -1, -1,  1, -1, ...
   3 |   5 | -1, -1,  0, -1,  1, -1, -1,  1, -1,  1, ...
   4 |   7 |  1,  1, -1,  0, -1, -1, -1,  1, -1,  1, ...
   5 |  11 | -1, -1,  1,  1,  0, -1, -1,  1, -1, -1, ...
   6 |  13 | -1,  1, -1, -1, -1,  0,  1, -1,  1,  1, ...
   7 |  17 |  1, -1, -1, -1, -1,  1,  0,  1, -1, -1, ...
   8 |  19 | -1,  1,  1, -1, -1, -1,  1,  0, -1, -1, ...
   9 |  23 |  1, -1, -1,  1,  1,  1, -1,  1,  0,  1, ...
  10 |  29 | -1, -1,  1,  1, -1,  1, -1, -1,  1,  0, ...
  ...
		

Links

• Wikipedia, Chebyshev's bias

Crossrefs

Cf. A215200.

Cf. A226523 (1st row and 1st column), A257834 (2nd row), A134323 (2nd column).

Programs

PARI

T(n,k) = kronecker(prime(n), prime(k))

Formula

T(n,k) = A215200(prime(n) + prime(k), prime(k)).