A345418 - OEIS

%I #19 Dec 30 2023 23:20:13

%S 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,

%T -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,

%U -15,10,-9,-8,-3,1,2,3,2,-1,1,-18,-10,6,10,7,4,-3,-4,-3,-1,1,1

%N 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.

%C 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.

%e The u table (A345417) begins:

%e [0, -1, -2, -3, -5, -6, -8, -9, -11, -14, -15, -18, -20, -21, -23, -26]

%e [1,  0,  2, -2,  4, -4,  6, -6,   8,  10, -10, -12,  14, -14,  16,  18]

%e [1, -1,  0,  3, -2, -5,  7,  4,  -9,   6,  -6,  15,  -8, -17,  19, -21]

%e [1,  1, -2,  0, -3,  2,  5, -8,  10,  -4,   9,  16,   6,  -6, -20, -15]

%e [1, -1,  1,  2,  0,  6, -3,  7,  -2,   8, -14, -10,  15,   4, -17, -24]

%e [1,  1,  2, -1, -5,  0,  4,  3,  -7,   9,  12, -17,  19,  10, -18,  -4]

%e [1, -1, -2, -2,  2, -3,  0,  9,  -4,  12,  11, -13, -12,  -5, -11,  25]

%e [1,  1, -1,  3, -4, -2, -8,  0,  -6,  -3, -13,   2,  13,  -9,   5,  14]

%e [1, -1,  2, -3,  1,  4,  3,  5,   0,  -5,  -4,  -8, -16,  15,  -2, -23]

%e [1, -1, -1,  1, -3, -4, -7,  2,   4,   0,  15, -14,  17,   3,  13,  11]

%e [1,  1,  1, -2,  5, -5, -6,  8,   3, -14,   0,   6,   4, -18,  -3,  12]

%e [1,  1, -2, -3,  3,  6,  6, -1,   5,  11,  -5,   0,  10,   7,  14, -10]

%e ...

%e The v table (this entry) begins:

%e [  1,   1,  1,  1,   1,   1,   1,   1,  1,   1,   1,  1,   1,  1,   1,  1]

%e [ -1,   1, -1,  1,  -1,   1,  -1,   1, -1,  -1,   1,  1,  -1,  1,  -1, -1]

%e [ -2,   2,  1, -2,   1,   2,  -2,  -1,  2,  -1,   1, -2,   1,  2,  -2,  2]

%e [ -3,  -2,  3,  1,   2,  -1,  -2,   3, -3,   1,  -2, -3,  -1,  1,   3,  2]

%e [ -5,   4, -2, -3,   1,  -5,   2,  -4,  1,  -3,   5,  3,  -4, -1,   4,  5]

%e [ -6,  -4, -5,  2,   6,   1,  -3,  -2,  4,  -4,  -5,  6,  -6, -3,   5,  1]

%e [ -8,   6,  7,  5,  -3,   4,   1,  -8,  3,  -7,  -6,  6,   5,  2,   4, -8]

%e [ -9,  -6,  4, -8,   7,   3,   9,   1,  5,   2,   8, -1,  -6,  4,  -2, -5]

%e [-11,   8, -9, 10,  -2,  -7,  -4,  -6,  1,   4,   3,  5,   9, -8,   1, 10]

%e [-14,  10,  6, -4,   8,   9,  12,  -3, -5,   1, -14, 11, -12, -2,  -8, -6]

%e [-15, -10, -6,  9, -14,  12,  11, -13, -4,  15,   1, -5,  -3, 13,   2, -7]

%e [-18, -12, 15, 16, -10, -17, -13,   2, -8, -14,   6,  1,  -9, -6, -11,  7]

%e ...

%Y Cf. A003989, A050873, A345415, A345416, A345417, A345419, A345420, A345421, A345422.

%K sign,tabl

%O 1,4

%A _N. J. A. Sloane_, Jun 19 2021