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[ 7, 7, 10, 19, 40, 40, 40, 40, 7, 8, 10, 26, 40, 40, 40, 40, 10, 10, 22, 40, 40, 40, 40, 40, 19, 26, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40...
[ 10, 11, 14, 28, 59, 59, 59, 59, 11, 13, 16, 40, 59, 59, 59, 59, 14, 16, 34, 59, 59, 59, 59, 59, 28, 40, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, ...
[ 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, 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...
[ 1, 1, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 3, 3, 3, 3, 1, 1, 1, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3...
[ 9, 9, 12, 24, 50, 50, 50, 50, 9, 11, 13, 33, 50, 50, 50, 50, 12, 13, 28, 50, 50, 50, 50, 50, 24, 33, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 5...
[ 2, 2, 3, 5, 12, 12, 12, 12, 2, 3, 3, 8, 12, 12, 12, 12, 3, 3, 7, 12, 12, 12, 12, 12, 5, 8, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, ...
[ 4, 4, 5, 10, 22, 22, 22, 22, 4, 5, 6, 15, 22, 22, 22, 22, 5, 6, 12, 22, 22, 22, 22, 22, 10, 15, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, ...
[ 7, 8, 10, 20, 42, 42, 42, 42, 8, 9, 11, 28, 42, 42, 42, 42, 10, 11, 24, 42, 42, 42, 42, 42, 20, 28, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42...
[ 3, 3, 7, 13, 15, 15, 15, 15, 3, 4, 7, 13, 14, 12, 12, 12, 7, 7, 13, 14, 12, 12, 12, 12, 13, 13, 14, 12, 12, 12, 12, 12, 15, 14, 12, 12, 12, 12, 12, 12, 15, 12, 12, 12, 12, 12, 12, 12, 15, 12, 12, 12, 12, 12, ...
[ 1, 1, 2, 4, 9, 9, 9, 9, 1, 2, 2, 6, 9, 9, 9, 9, 2, 2, 5, 9, 9, 9, 9, 9, 4, 6, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9...
[ 14, 14, 19, 38, 79, 79, 79, 79, 14, 17, 21, 53, 79, 79, 79, 79, 19, 21, 45, 79, 79, 79, 79, 79, 38, 53, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, ...
[ 1, 1, 2, 4, 8, 8, 8, 8, 1, 2, 2, 5, 8, 8, 8, 8, 2, 2, 4, 8, 8, 8, 8, 8, 4, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8...
[ 1, 1, 1, 1, 2, 3, 3, 3, 1, 1, 1, 2, 3, 3, 3, 3, 1, 1, 1, 3, 3, 3, 3, 3, 1, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3...
[ 6, 6, 6, 7, 10, 15, 22, 34, 6, 7, 8, 11, 14, 16, 21, 30, 6, 8, 10, 12, 17, 25, 36, 54, 7, 11, 12, 16, 21, 30, 42, 62, 10, 14, 17, 21, 28, 38, 52, 76, 15, 16, 25, 30, 38, 50, 68, 95, 22, 21, 36, 42, 52, 68, 90...
[ 3, 4, 5, 9, 20, 20, 20, 20, 4, 4, 5, 13, 20, 20, 20, 20, 5, 5, 11, 20, 20, 20, 20, 20, 9, 13, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20...
[ 7, 7, 13, 24, 26, 31, 31, 31, 7, 12, 16, 21, 31, 31, 31, 31, 13, 16, 17, 31, 31, 31, 31, 31, 24, 21, 31, 31, 31, 31, 31, 31, 26, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 3...
[ 2, 2, 4, 7, 16, 16, 16, 16, 2, 4, 4, 11, 16, 16, 16, 16, 4, 4, 9, 16, 16, 16, 16, 16, 7, 11, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16,...
[ 2, 2, 2, 5, 10, 10, 10, 10, 2, 2, 3, 7, 10, 10, 10, 10, 2, 3, 6, 10, 10, 10, 10, 10, 5, 7, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, ...
[ 3, 3, 5, 9, 21, 21, 21, 21, 3, 5, 5, 14, 21, 21, 21, 21, 5, 5, 12, 21, 21, 21, 21, 21, 9, 14, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21...
[ 9, 9, 13, 15, 38, 38, 38, 38, 9, 10, 13, 21, 38, 38, 50, 38, 13, 13, 16, 38, 38, 50, 50, 50, 15, 21, 38, 50, 38, 50, 38, 50, 38, 38, 38, 38, 50, 38, 50, 50, 38, 38, 38, 50, 38, 50, 50, 38, 38, 38, 38, 38, 38, 3...
[ 4, 5, 4, 4, 4, 4, 5, 6, 5, 4, 4, 4, 4, 4, 5, 6, 4, 4, 4, 4, 4, 5, 6, 7, 4, 4, 4, 5, 5, 6, 7, 8, 4, 4, 4, 5, 7, 8, 10, 11, 4, 4, 5, 6, 8, 11, 13, 16, 5, 5, 6, 7, 10, 13, 17, 21, 6, 6, 7, 8, 11, 1...
[ 11, 11, 15, 29, 61, 61, 61, 61, 11, 13, 16, 41, 61, 61, 61, 61, 15, 16, 35, 61, 61, 61, 61, 61, 29, 41, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, ...
[ 2, 2, 3, 6, 13, 13, 13, 13, 2, 3, 3, 9, 13, 13, 13, 13, 3, 3, 7, 13, 13, 13, 13, 13, 6, 9, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, ...
[ 5, 5, 10, 8, 11, 10, 12, 13, 5, 7, 9, 8, 10, 9, 11, 12, 10, 9, 10, 9, 10, 10, 11, 12, 8, 8, 9, 9, 11, 11, 12, 13, 11, 10, 10, 11, 8, 13, 10, 12, 10, 9, 10, 11, 13, 11, 13, 19, 12, 11, 11, 12, 10, 13, 20, 19...
[ 1, 1, 1, 2, 4, 4, 4, 4, 1, 1, 1, 3, 4, 4, 4, 4, 1, 1, 2, 4, 4, 4, 4, 4, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4...
[ 2, 3, 4, 7, 16, 16, 16, 16, 3, 4, 4, 11, 16, 16, 16, 16, 4, 4, 9, 16, 16, 16, 16, 16, 7, 11, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16,...
[ 4, 5, 4, 4, 6, 6, 6, 6, 5, 4, 4, 4, 6, 6, 6, 6, 4, 4, 4, 6, 6, 6, 6, 7, 4, 4, 6, 6, 6, 6, 7, 8, 6, 6, 6, 6, 7, 8, 10, 11, 6, 6, 6, 6, 8, 11, 13, 16, 6, 6, 6, 7, 10, 13, 17, 21, 6, 6, 7, 8, 11, 1...
[ 5, 6, 5, 5, 6, 6, 6, 7, 6, 5, 5, 5, 6, 6, 6, 7, 5, 5, 5, 6, 6, 6, 7, 8, 5, 5, 6, 6, 7, 8, 9, 11, 6, 6, 6, 7, 9, 10, 12, 14, 6, 6, 6, 8, 10, 13, 16, 20, 6, 6, 7, 9, 12, 16, 21, 27, 7, 7, 8, 11, 14,...
[ 3, 3, 5, 9, 13, 15, 15, 15, 3, 4, 6, 11, 14, 12, 12, 12, 5, 6, 9, 14, 12, 12, 12, 12, 9, 11, 14, 12, 12, 12, 12, 12, 13, 14, 12, 12, 12, 12, 12, 12, 15, 12, 12, 12, 12, 12, 12, 12, 15, 12, 12, 12, 12, 12, 12,...
[ 2, 2, 3, 6, 12, 12, 12, 12, 2, 3, 3, 8, 12, 12, 12, 12, 3, 3, 7, 12, 12, 12, 12, 12, 6, 8, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, ...
[ 3, 3, 6, 11, 17, 20, 17, 17, 3, 6, 8, 13, 15, 15, 15, 15, 6, 8, 11, 15, 15, 15, 15, 15, 11, 13, 15, 15, 15, 15, 15, 15, 17, 15, 15, 15, 15, 15, 15, 15, 20, 15, 15, 15, 15, 15, 15, 15, 17, 15, 15, 15, 15, 15, ...
[ 3, 3, 4, 8, 16, 16, 16, 16, 3, 3, 4, 11, 16, 16, 16, 16, 4, 4, 9, 16, 16, 16, 16, 16, 8, 11, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16,...
[ 11, 10, 11, 14, 18, 20, 25, 32, 10, 12, 12, 14, 14, 17, 23, 28, 11, 12, 14, 14, 17, 20, 30, 35, 14, 14, 14, 18, 20, 26, 35, 43, 18, 14, 17, 20, 26, 32, 43, 50, 20, 17, 20, 26, 32, 41, 50, 50, 25, 23, 30, 35, 43, ...
[ 1, 1, 2, 5, 8, 8, 8, 8, 1, 2, 3, 7, 8, 8, 8, 8, 2, 3, 6, 8, 8, 8, 8, 8, 5, 7, 8, 8, 8, 8, 8, 8, 8, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8...
[ 9, 9, 13, 15, 38, 38, 38, 38, 9, 11, 13, 21, 38, 38, 38, 38, 13, 13, 21, 38, 50, 50, 50, 50, 21, 21, 38, 50, 38, 38, 38, 38, 38, 38, 38, 38, 38, 50, 50, 38, 38, 50, 38, 38, 38, 38, 38, 38, 38, 50, 38, 38, 38, 3...
[ 9, 9, 12, 15, 38, 38, 38, 38, 9, 11, 13, 21, 38, 38, 38, 38, 13, 13, 21, 38, 38, 38, 38, 38, 21, 21, 38, 38, 38, 38, 38, 38, 38, 38, 38, 50, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 50, 38, 38, 38, 38, 38, 3...
[ 2, 2, 3, 7, 15, 15, 15, 15, 2, 3, 4, 10, 15, 15, 15, 15, 3, 4, 8, 15, 15, 15, 15, 15, 7, 10, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15,...
[ 2, 2, 10, 10, 10, 10, 10, 10, 2, 10, 10, 10, 2, 10, 10, 10, 5, 2, 10, 3, 7, 10, 10, 10, 2, 7, 6, 5, 10, 10, 10, 10, 3, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, ...
[ 3, 3, 5, 8, 19, 19, 19, 19, 3, 4, 4, 13, 19, 19, 19, 19, 5, 4, 11, 19, 19, 19, 19, 19, 8, 13, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19...
[ 1, 1, 1, 3, 6, 6, 6, 6, 1, 1, 2, 4, 6, 6, 6, 6, 1, 2, 3, 6, 6, 6, 6, 6, 3, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6...
[ 3, 4, 6, 10, 13, 13, 13, 13, 4, 5, 6, 13, 13, 13, 13, 8, 6, 6, 8, 13, 13, 13, 8, 8, 10, 13, 13, 13, 13, 8, 8, 8, 13, 13, 13, 13, 8, 8, 8, 8, 13, 13, 13, 13, 8, 8, 8, 8, 13, 8, 8, 8, 8, 8, 8, 8, 13, 8, ...
[ 7, 7, 13, 24, 20, 20, 17, 17, 7, 12, 16, 14, 14, 12, 12, 12, 13, 16, 14, 14, 12, 12, 12, 12, 24, 14, 14, 12, 12, 12, 12, 12, 20, 14, 12, 12, 12, 12, 12, 12, 20, 12, 12, 12, 12, 12, 12, 12, 17, 12, 12, 12, 12, 1...
[ 13, 13, 17, 27, 20, 20, 17, 17, 13, 14, 17, 14, 14, 12, 12, 12, 17, 17, 14, 14, 12, 12, 12, 12, 27, 14, 14, 12, 12, 12, 12, 12, 20, 14, 12, 12, 12, 12, 12, 12, 20, 12, 12, 12, 12, 12, 12, 12, 17, 12, 12, 12, 12, ...
[ 5, 6, 5, 5, 5, 5, 6, 7, 6, 5, 5, 5, 5, 5, 6, 7, 5, 5, 5, 5, 6, 6, 7, 8, 5, 5, 5, 6, 7, 8, 9, 11, 5, 5, 6, 7, 9, 10, 12, 14, 5, 5, 6, 8, 10, 13, 16, 20, 6, 6, 7, 9, 12, 16, 21, 27, 7, 7, 8, 11, 14,...
[ 5, 6, 8, 15, 32, 32, 32, 32, 6, 7, 8, 21, 32, 32, 32, 32, 8, 8, 18, 32, 32, 32, 32, 32, 15, 21, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, ...
[ 17, 18, 24, 47, 99, 99, 99, 99, 18, 21, 26, 66, 99, 99, 99, 99, 24, 26, 56, 99, 99, 99, 99, 99, 47, 66, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, ...
[ 1, 1, 2, 4, 8, 8, 8, 8, 1, 2, 2, 5, 8, 8, 8, 8, 2, 2, 5, 8, 8, 8, 8, 8, 4, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8...
[ 3, 3, 5, 9, 22, 22, 22, 22, 3, 5, 5, 15, 22, 22, 22, 22, 5, 5, 12, 22, 22, 22, 22, 22, 9, 15, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22...
[ 2, 2, 3, 6, 14, 14, 14, 14, 2, 3, 3, 9, 14, 14, 14, 14, 3, 3, 8, 14, 14, 14, 14, 14, 6, 9, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, ...
[ 19, 20, 27, 52, 110, 110, 110, 110, 20, 23, 29, 73, 110, 110, 110, 110, 27, 29, 62, 110, 110, 110, 110, 110, 52, 73, 110, 110, 110, 110, 110, 110, 110, 110, 110, 110, 110, 110, 110, 110, 110, 110, 110, 110, 110, 110, 110, ...
[ 14, 15, 20, 39, 81, 81, 81, 81, 15, 17, 21, 54, 81, 81, 81, 81, 20, 21, 46, 81, 81, 81, 81, 81, 39, 54, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, ...
[ 3, 3, 5, 7, 11, 11, 11, 12, 3, 4, 6, 8, 11, 11, 12, 12, 5, 6, 8, 11, 11, 11, 12, 12, 7, 8, 11, 11, 11, 11, 12, 12, 11, 11, 11, 11, 11, 12, 12, 12, 11, 11, 11, 12, 12, 12, 12, 12, 11, 12, 12, 12, 12, 12, 12, ...
[ 3, 4, 5, 6, 8, 8, 8, 8, 4, 5, 6, 6, 8, 8, 8, 8, 5, 6, 6, 8, 8, 8, 8, 8, 5, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 20, 8, 8, 8, 8, 8, 8, ...
[ 3, 4, 6, 6, 8, 8, 8, 8, 4, 5, 6, 7, 8, 8, 8, 8, 6, 6, 7, 8, 8, 8, 8, 8, 8, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8...
[ 1, 1, 1, 2, 3, 3, 3, 3, 1, 1, 1, 2, 3, 3, 3, 3, 1, 1, 2, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3...
[ 12, 13, 17, 33, 69, 69, 69, 69, 13, 15, 18, 46, 69, 69, 69, 69, 17, 18, 39, 69, 69, 69, 69, 69, 33, 46, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, ...
[ 7, 9, 19, 34, 20, 20, 17, 17, 9, 12, 19, 14, 14, 12, 12, 12, 19, 19, 14, 14, 12, 12, 12, 12, 34, 14, 14, 12, 12, 12, 12, 12, 20, 14, 12, 12, 12, 12, 12, 12, 20, 12, 12, 12, 12, 12, 12, 12, 17, 12, 12, 12, 12, 1...
[ 7, 7, 10, 20, 28, 43, 43, 43, 7, 9, 11, 28, 43, 43, 43, 43, 10, 11, 24, 43, 43, 43, 43, 43, 20, 28, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43...
[ 3, 3, 3, 4, 5, 7, 11, 17, 3, 3, 4, 5, 7, 8, 11, 15, 3, 4, 5, 6, 9, 12, 18, 27, 4, 5, 6, 8, 11, 15, 21, 31, 5, 7, 9, 11, 14, 19, 26, 38, 7, 8, 12, 15, 19, 25, 34, 48, 11, 11, 18, 21, 26, 34, 45, 62, 17, ...
[ 7, 8, 11, 21, 44, 44, 44, 44, 8, 9, 11, 29, 44, 44, 44, 44, 11, 11, 25, 44, 44, 44, 44, 44, 21, 29, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44...
[ 7, 7, 11, 19, 46, 46, 46, 46, 7, 10, 10, 31, 46, 46, 46, 46, 11, 10, 26, 46, 46, 46, 46, 46, 19, 31, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 4...
[ 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 3, 4, 4, 5, 3, 3, 3, 3, 4, 5, 5, 6, 3, 3, 3, 4, 5, 6, 7, 8, 3, 3, 4, 5, 6, 8, 10, 12, 3, 4, 4, 5, 7, 10, 13, 16, 4, 4, 5, 6, 8, 12, ...
[ 1, 1, 1, 1, 4, 4, 4, 4, 1, 1, 2, 2, 4, 4, 4, 4, 1, 2, 3, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4...
[ 4, 5, 6, 12, 26, 26, 26, 26, 5, 5, 7, 17, 26, 26, 26, 26, 6, 7, 15, 26, 26, 26, 26, 26, 12, 17, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, ...
[ 13, 13, 13, 14, 20, 30, 45, 68, 13, 14, 16, 22, 27, 32, 42, 60, 13, 16, 19, 25, 34, 50, 73, 108, 14, 22, 25, 32, 42, 59, 85, 125, 20, 27, 34, 42, 55, 75, 105, 151, 30, 32, 50, 59, 75, 99, 135, 190, 45, 42, 73, 85, ...
[ 14, 15, 24, 40, 97, 97, 97, 97, 15, 22, 22, 66, 97, 97, 97, 97, 24, 22, 54, 97, 97, 97, 97, 97, 40, 66, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, ...
[ 8, 8, 12, 21, 51, 51, 51, 51, 8, 11, 11, 34, 51, 51, 51, 51, 12, 11, 28, 51, 51, 51, 51, 51, 21, 34, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 5...
[ 4, 5, 6, 13, 27, 27, 27, 27, 5, 5, 7, 18, 27, 27, 27, 27, 6, 7, 15, 27, 27, 27, 27, 27, 13, 18, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, ...
[ 13, 11, 13, 16, 20, 20, 17, 17, 11, 14, 14, 14, 14, 12, 12, 12, 13, 14, 14, 14, 12, 12, 12, 12, 16, 14, 14, 12, 12, 12, 12, 12, 20, 14, 12, 12, 12, 12, 12, 12, 20, 12, 12, 12, 12, 12, 12, 12, 17, 12, 12, 12, 12, ...
[ 27, 29, 39, 76, 159, 159, 159, 159, 29, 34, 42, 106, 159, 159, 159, 159, 39, 42, 90, 159, 159, 159, 159, 159, 76, 106, 159, 159, 159, 159, 159, 159, 159, 159, 159, 159, 159, 159, 159, 159, 159, 159, 159, 159, 159, 159, 159, ...
[ 28, 30, 40, 78, 164, 164, 164, 164, 30, 35, 43, 110, 164, 164, 164, 164, 40, 43, 93, 164, 164, 164, 164, 164, 78, 110, 164, 164, 164, 164, 164, 164, 164, 164, 164, 164, 164, 164, 164, 164, 164, 164, 164, 164, 164, 164, 164, ...
[ 6, 6, 8, 16, 34, 34, 34, 34, 6, 7, 9, 22, 34, 34, 34, 34, 8, 9, 19, 34, 34, 34, 34, 34, 16, 22, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, ...
[ 3, 4, 10, 15, 15, 15, 15, 15, 4, 5, 10, 14, 14, 12, 12, 12, 10, 10, 14, 14, 14, 17, 17, 17, 15, 14, 14, 14, 17, 17, 17, 17, 15, 14, 14, 17, 17, 17, 17, 17, 15, 12, 17, 17, 17, 17, 17, 17, 15, 12, 17, 17, 17, 17...
[ 1, 3, 3, 4, 8, 8, 8, 8, 3, 3, 3, 5, 8, 8, 8, 8, 3, 3, 4, 8, 8, 8, 8, 8, 4, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 12, 8, 8, 8, 8, 8, 9, 12, 15, 8, 8, 8, 8, 8, 12, 15...
[ 3, 3, 5, 5, 6, 5, 6, 7, 3, 4, 5, 4, 5, 5, 6, 6, 5, 5, 5, 5, 5, 5, 6, 6, 5, 4, 5, 4, 6, 6, 6, 7, 6, 5, 5, 6, 4, 7, 5, 7, 5, 5, 5, 6, 7, 6, 7, 10, 6, 6, 6, 6, 5, 7, 11, 10, 7, 6, 6, 7, 7, 10, 10...
[ 17, 15, 17, 21, 20, 26, 38, 48, 15, 19, 18, 17, 20, 26, 35, 43, 17, 18, 20, 22, 26, 30, 46, 53, 21, 17, 22, 28, 30, 39, 53, 64, 20, 20, 26, 30, 39, 48, 64, 64, 26, 26, 30, 39, 48, 63, 64, 64, 38, 35, 46, 53, 64, ...
[ 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2...
[ 7, 7, 13, 24, 26, 32, 32, 32, 7, 12, 16, 21, 32, 32, 32, 32, 13, 16, 17, 32, 32, 32, 32, 32, 24, 21, 32, 32, 32, 32, 32, 32, 26, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 3...
[ 8, 8, 8, 9, 13, 19, 28, 43, 8, 9, 10, 14, 17, 20, 27, 38, 8, 10, 12, 16, 22, 31, 46, 68, 9, 14, 16, 20, 27, 37, 53, 78, 13, 17, 22, 27, 35, 47, 66, 95, 19, 20, 31, 37, 47, 62, 85, 119, 28, 27, 46, 53, 66, 85, ...
[ 1, 1, 2, 4, 6, 7, 7, 7, 1, 2, 3, 5, 7, 7, 7, 7, 2, 3, 4, 7, 7, 7, 7, 7, 4, 5, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7...
[ 7, 8, 12, 20, 48, 48, 48, 48, 8, 11, 11, 33, 48, 48, 48, 48, 12, 11, 27, 48, 48, 48, 48, 48, 20, 33, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 4...
[ 8, 9, 14, 23, 56, 56, 56, 56, 9, 13, 13, 38, 56, 56, 56, 56, 14, 13, 31, 56, 56, 56, 56, 56, 23, 38, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 5...
[ 11, 11, 15, 25, 47, 50, 60, 47, 11, 14, 17, 40, 49, 51, 58, 49, 15, 17, 32, 49, 49, 49, 60, 50, 25, 32, 47, 51, 52, 49, 49, 47, 52, 49, 49, 60, 53, 49, 49, 50, 49, 47, 49, 47, 61, 47, 47, 49, 60, 47, 50, 60, 49, ...
[ 3, 4, 6, 10, 13, 13, 13, 13, 4, 5, 6, 13, 14, 13, 13, 13, 6, 6, 10, 13, 13, 13, 13, 13, 10, 14, 13, 13, 13, 13, 13, 13, 13, 13, 15, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, ...
[ 1, 4, 3, 4, 8, 8, 8, 8, 4, 3, 3, 5, 8, 8, 8, 8, 3, 3, 4, 8, 8, 8, 8, 8, 4, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 8, 8, 8, 8, 8, 9, 12, 15, 8, 8, 8, 8, 9, 12, 15, 20, 8, 8, 8, 8, 10, 15, ...
[ 10, 10, 14, 28, 59, 59, 59, 59, 10, 12, 15, 39, 59, 59, 59, 59, 14, 15, 33, 59, 59, 59, 59, 59, 28, 39, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, ...
[ 2, 2, 3, 6, 14, 14, 14, 14, 2, 3, 4, 9, 14, 14, 14, 14, 3, 4, 8, 14, 14, 14, 14, 14, 6, 9, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, ...
[ 3, 4, 6, 10, 13, 8, 13, 13, 4, 5, 6, 8, 8, 8, 13, 13, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, ...
[ 3, 3, 5, 9, 13, 15, 15, 15, 3, 4, 6, 10, 14, 12, 12, 12, 5, 6, 9, 14, 12, 12, 12, 12, 9, 10, 14, 12, 12, 12, 12, 12, 13, 14, 12, 12, 12, 12, 12, 12, 15, 12, 12, 12, 12, 12, 12, 12, 15, 12, 12, 12, 12, 12, 12,...
[ 5, 5, 7, 14, 30, 30, 30, 30, 5, 6, 8, 20, 30, 30, 30, 30, 7, 8, 17, 30, 30, 30, 30, 30, 14, 20, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, ...
[ 4, 4, 6, 11, 24, 24, 24, 24, 4, 5, 6, 16, 24, 24, 24, 24, 6, 6, 13, 24, 24, 24, 24, 24, 11, 16, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, ...
[ 9, 8, 9, 11, 14, 17, 17, 17, 8, 10, 9, 11, 14, 12, 12, 12, 9, 9, 13, 14, 12, 12, 12, 12, 11, 11, 14, 12, 12, 12, 12, 12, 14, 14, 12, 12, 12, 12, 12, 12, 17, 12, 12, 12, 12, 12, 12, 12, 17, 12, 12, 12, 12, 12, ...
[ 2, 2, 3, 5, 11, 11, 11, 11, 2, 3, 3, 8, 11, 11, 11, 11, 3, 3, 6, 11, 11, 11, 11, 11, 5, 8, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, ...
[ 11, 16, 20, 24, 27, 27, 27, 27, 16, 15, 15, 24, 27, 27, 27, 27, 20, 15, 20, 27, 27, 27, 27, 27, 24, 24, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, ...
[ 3, 3, 7, 13, 15, 15, 15, 15, 3, 5, 7, 13, 14, 12, 12, 12, 7, 7, 13, 14, 12, 14, 17, 17, 13, 13, 14, 12, 14, 17, 17, 17, 15, 14, 12, 14, 17, 17, 17, 17, 15, 12, 14, 17, 17, 17, 17, 17, 15, 12, 17, 17, 17, 17, ...
[ 10, 10, 14, 27, 44, 45, 55, 44, 10, 13, 16, 38, 52, 52, 52, 43, 12, 16, 29, 55, 55, 44, 44, 44, 19, 32, 55, 53, 43, 44, 53, 44, 46, 44, 54, 52, 44, 44, 45, 43, 52, 52, 43, 45, 44, 55, 44, 56, 52, 43, 44, 52, 55, ...
[ 2, 1, 1, 2, 2, 4, 5, 6, 1, 1, 1, 2, 3, 6, 6, 6, 1, 1, 2, 2, 4, 6, 7, 6, 1, 2, 2, 3, 5, 9, 8, 6, 2, 2, 4, 6, 7, 11, 10, 8, 2, 4, 6, 6, 8, 10, 11, 9, 5, 6, 8, 9, 10, 12, 12, 10, 7, 9, 10, 10, 11, ...
[ 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 3, 4, 2, 2, 2, 2, 3, 4, 4, 5, 2, 2, 2, 2, 3, 4, 6, 7, 2, 2, 2, 3, 4, 5, 7, 9...
[ 6, 7, 9, 18, 38, 38, 38, 38, 7, 8, 10, 25, 38, 38, 38, 38, 9, 10, 21, 38, 38, 38, 38, 38, 18, 25, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, ...
[ 8, 8, 11, 22, 46, 46, 46, 46, 8, 10, 12, 30, 46, 46, 46, 46, 11, 12, 26, 46, 46, 46, 46, 46, 22, 30, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 4...
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DocQT - Improving Document Forgery Localization Robustness via Diverse JPEG Quantization-Tables

Repository containing JPEG quantization tables used for the Real-QT protocol. Dataset name: DocQT. Only header-extracted quantization matrices are provided.

Paper

Citation

If you use DocQT, this quantization-table repository, or build upon our article, please cite our paper:

@misc{ronfleuxcorail2026docqt,
    title={DocQT: Improving Document Forgery Localization Robustness via Diverse JPEG Quantization Tables},
    author={Kylian Ronfleux-Corail and Guillaume Bernard and Mickael Coustaty and Nicolas Sidere},
    year={2026},
    eprint={2605.19688},
    archivePrefix={arXiv},
    primaryClass={cs.CV},
    doi={10.48550/arXiv.2605.19688},
    url={https://arxiv.org/abs/2605.19688}
}

Available quantization tables

  • Luminance tables: quantification_luminance.json
    • Number of tables: 859
  • Chrominance tables: quantification_chrominance.json
    • Number of tables: 294

File format

Both files are stored in JSON for better portability and long-term compatibility.

  • Root object: a list of quantization tables
  • One table: a flat list of 64 integers
  • Table values: integers

In other words, each file follows this structure:

  • list[table]
  • table = list[64 integer values]

Example: use quantization tables with Pillow JPEG compression

import json
from pathlib import Path
from PIL import Image


def load_quantization_tables(base_dir: str = ".") -> tuple[list, list]:
    base_path = Path(base_dir)

    luminance_tables = json.loads(
        (base_path / "quantification_luminance.json").read_text(encoding="utf-8")
    )
    chrominance_tables = json.loads(
        (base_path / "quantification_chrominance.json").read_text(encoding="utf-8")
    )

    return luminance_tables, chrominance_tables

luminance_tables, chrominance_tables = load_quantization_tables(".")

# Choose the table indices you want to use for compression.
luma_idx = 0
chroma_idx = 0

luma_qtable = luminance_tables[luma_idx]
chroma_qtable = chrominance_tables[chroma_idx]

with Image.open("input.png") as image:
    image = image.convert("RGB")
    image.save(
        "output_custom_qtables.jpg",
        format="JPEG",
        qtables=[luma_qtable, chroma_qtable],
        subsampling="4:2:0",
        optimize=True,
    )

# Note: avoid passing a quality value if you want to keep your custom qtables as-is.
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