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A345418
Table read by upward antidiagonals: Given m, n >= 1, write gcd(prime(m),prime(n)) as d = u*prime(m)+v*prime(n) where u, v are minimal; T(m,n) = v.
3
1, -1, 1, -2, 1, 1, -3, 2, -1, 1, -5, -2, 1, 1, 1, -6, 4, 3, -2, -1, 1, -8, -4, -2, 1, 1, 1, 1, -9, 6, -5, -3, 2, 2, -1, 1, -11, -6, 7, 2, 1, -1, -2, 1, 1, -14, 8, 4, 5, 6, -5, -2, -1, -1, 1, -15, 10, -9, -8, -3, 1, 2, 3, 2, -1, 1, -18, -10, 6, 10, 7, 4, -3, -4, -3, -1, 1, 1
OFFSET
1,4
COMMENTS
The gcd is 1 unless m=n when it is m; u is given in A345417. Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when m=n. If we ignore the diagonal, the v table is the transpose of the u table.
EXAMPLE
The u table (A345417) begins:
[0, -1, -2, -3, -5, -6, -8, -9, -11, -14, -15, -18, -20, -21, -23, -26]
[1, 0, 2, -2, 4, -4, 6, -6, 8, 10, -10, -12, 14, -14, 16, 18]
[1, -1, 0, 3, -2, -5, 7, 4, -9, 6, -6, 15, -8, -17, 19, -21]
[1, 1, -2, 0, -3, 2, 5, -8, 10, -4, 9, 16, 6, -6, -20, -15]
[1, -1, 1, 2, 0, 6, -3, 7, -2, 8, -14, -10, 15, 4, -17, -24]
[1, 1, 2, -1, -5, 0, 4, 3, -7, 9, 12, -17, 19, 10, -18, -4]
[1, -1, -2, -2, 2, -3, 0, 9, -4, 12, 11, -13, -12, -5, -11, 25]
[1, 1, -1, 3, -4, -2, -8, 0, -6, -3, -13, 2, 13, -9, 5, 14]
[1, -1, 2, -3, 1, 4, 3, 5, 0, -5, -4, -8, -16, 15, -2, -23]
[1, -1, -1, 1, -3, -4, -7, 2, 4, 0, 15, -14, 17, 3, 13, 11]
[1, 1, 1, -2, 5, -5, -6, 8, 3, -14, 0, 6, 4, -18, -3, 12]
[1, 1, -2, -3, 3, 6, 6, -1, 5, 11, -5, 0, 10, 7, 14, -10]
...
The v table (this entry) begins:
[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
[ -1, 1, -1, 1, -1, 1, -1, 1, -1, -1, 1, 1, -1, 1, -1, -1]
[ -2, 2, 1, -2, 1, 2, -2, -1, 2, -1, 1, -2, 1, 2, -2, 2]
[ -3, -2, 3, 1, 2, -1, -2, 3, -3, 1, -2, -3, -1, 1, 3, 2]
[ -5, 4, -2, -3, 1, -5, 2, -4, 1, -3, 5, 3, -4, -1, 4, 5]
[ -6, -4, -5, 2, 6, 1, -3, -2, 4, -4, -5, 6, -6, -3, 5, 1]
[ -8, 6, 7, 5, -3, 4, 1, -8, 3, -7, -6, 6, 5, 2, 4, -8]
[ -9, -6, 4, -8, 7, 3, 9, 1, 5, 2, 8, -1, -6, 4, -2, -5]
[-11, 8, -9, 10, -2, -7, -4, -6, 1, 4, 3, 5, 9, -8, 1, 10]
[-14, 10, 6, -4, 8, 9, 12, -3, -5, 1, -14, 11, -12, -2, -8, -6]
[-15, -10, -6, 9, -14, 12, 11, -13, -4, 15, 1, -5, -3, 13, 2, -7]
[-18, -12, 15, 16, -10, -17, -13, 2, -8, -14, 6, 1, -9, -6, -11, 7]
...
KEYWORD
sign,tabl
AUTHOR
N. J. A. Sloane, Jun 19 2021
STATUS
approved