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A323377
Square array read by ascending antidiagonals: T(n,k) = Kronecker(prime(n)/prime(k)), n, k >= 1.
0
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
1,1
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.
FORMULA
T(n,k) = A215200(prime(n) + prime(k), prime(k)).
EXAMPLE
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, ...
...
PROG
(PARI) T(n, k) = kronecker(prime(n), prime(k))
CROSSREFS
Cf. A215200.
Cf. A226523 (1st row and 1st column), A257834 (2nd row), A134323 (2nd column).
Sequence in context: A022930 A285596 A257680 * A068344 A161382 A138886
KEYWORD
sign,tabl
AUTHOR
Jianing Song, Jan 12 2019
STATUS
approved