「和の２乗 = ３乗の和」となる組合せ(1)
http://6626.teacup.com/shochandas/bbs/12272
において、
1≦s2≦s3≦s4≦･･･≦s(n-1)≦n
なる自然数の組[1,s2,s3,s4,･･･,s(n-1),n]
において関係式
(1+s2+s3+s4+･･･+s(n-1)+n)^2=1^3+s2^3+s3^3+s4^3+･･･+s(n-1)^3+n^3
が成立する組合せ
を発見して欲しいとあり、n が2以上12以下の場合についてすべて求めてみた。
出力結果
[[1, 2]]
[[1, 2, 3]]
[[1, 2, 2, 4], [1, 2, 3, 4]]
[[1, 2, 2, 3, 5], [1, 2, 3, 4, 5]]
[[1, 2, 2, 3, 4, 6], [1, 2, 2, 4, 4, 6], [1, 2, 3, 4, 5, 6], [1, 4, 4, 4, 6, 6]]
[[1, 1, 3, 4, 4, 5, 7], [1, 1, 4, 5, 5, 5, 7], [1, 2, 2, 3, 3, 3, 7], [1, 2, 2, 3, 4, 5, 7], [1, 2, 3, 4, 5, 6, 7], [1, 3, 5, 5, 5, 7, 7], [1, 4, 4, 5, 6, 7, 7], [1, 5, 5, 6, 6, 7, 7]]
[[1, 1, 2, 4, 4, 5, 5, 8], [1, 1, 2, 4, 5, 5, 5, 8], [1, 1, 3, 3, 3, 4, 5, 8], [1, 1, 4, 5, 5, 5, 7, 8], [1, 2, 2, 2, 4, 4, 4, 8], [1, 2, 2, 3, 4, 4, 6, 8], [1, 2, 2, 3, 4, 5, 6, 8], [1, 2, 2, 4, 4, 6, 6, 8], [1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 4, 5, 5, 7, 7, 8], [1, 2, 5, 6, 6, 7, 7, 8], [1, 3, 3, 3, 3, 5, 7, 8], [1, 3, 3, 3, 6, 6, 7, 8], [1, 3, 4, 4, 4, 5, 8, 8], [1, 3, 4, 4, 5, 6, 8, 8], [1, 3, 4, 6, 6, 6, 8, 8], [1, 3, 4, 6, 7, 7, 7, 8], [1, 4, 5, 5, 7, 7, 8, 8], [1, 4, 5, 6, 7, 7, 8, 8], [1, 5, 5, 7, 7, 7, 8, 8], [1, 6, 6, 6, 6, 8, 8, 8]]
[[1, 1, 1, 4, 5, 5, 5, 5, 9], [1, 1, 2, 3, 4, 5, 5, 6, 9], [1, 1, 2, 4, 5, 5, 5, 7, 9], [1, 1, 2, 5, 5, 6, 6, 7, 9], [1, 1, 3, 3, 3, 3, 5, 5, 9], [1, 1, 3, 3, 3, 4, 4, 6, 9], [1, 1, 3, 3, 3, 5, 6, 6, 9], [1, 1, 3, 4, 5, 6, 7, 7, 9], [1, 1, 4, 5, 5, 5, 7, 8, 9], [1, 1, 4, 5, 5, 6, 7, 8, 9], [1, 1, 5, 5, 5, 7, 7, 8, 9], [1, 1, 6, 6, 7, 7, 7, 8, 9], [1, 2, 2, 2, 3, 5, 5, 5, 9], [1, 2, 2, 3, 3, 4, 6, 6, 9], [1, 2, 2, 3, 4, 4, 5, 7, 9], [1, 2, 2, 3, 4, 5, 6, 7, 9], [1, 2, 2, 3, 6, 6, 6, 7, 9], [1, 2, 2, 5, 5, 5, 6, 8, 9], [1, 2, 2, 5, 5, 7, 7, 7, 9], [1, 2, 2, 5, 6, 7, 7, 7, 9], [1, 2, 3, 3, 3, 3, 3, 3, 9], [1, 2, 3, 3, 3, 3, 3, 6, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 6, 6, 7, 8, 9], [1, 2, 3, 5, 6, 7, 7, 8, 9], [1, 2, 4, 4, 4, 4, 7, 8, 9], [1, 2, 4, 5, 6, 6, 6, 9, 9], [1, 2, 4, 6, 6, 7, 8, 8, 9], [1, 2, 5, 5, 5, 6, 7, 9, 9], [1, 2, 5, 7, 7, 7, 8, 8, 9], [1, 3, 3, 3, 3, 5, 5, 8, 9], [1, 3, 3, 4, 6, 6, 8, 8, 9], [1, 3, 3, 5, 5, 7, 8, 8, 9], [1, 3, 4, 5, 7, 7, 7, 9, 9], [1, 3, 4, 6, 7, 8, 8, 8, 9], [1, 3, 5, 6, 7, 7, 8, 9, 9], [1, 3, 6, 7, 8, 8, 8, 8, 9], [1, 4, 4, 4, 4, 7, 7, 9, 9], [1, 4, 4, 7, 8, 8, 8, 8, 9], [1, 4, 5, 6, 6, 6, 9, 9, 9], [1, 5, 5, 5, 7, 7, 9, 9, 9], [1, 5, 6, 6, 7, 8, 9, 9, 9]]
[[1, 1, 1, 4, 4, 4, 4, 5, 6, 10], [1, 1, 1, 4, 4, 5, 5, 5, 7, 10], [1, 1, 1, 4, 6, 6, 6, 7, 7, 10], [1, 1, 2, 2, 4, 4, 5, 5, 5, 10], [1, 1, 2, 2, 4, 5, 5, 6, 6, 10], [1, 1, 2, 3, 4, 4, 5, 6, 7, 10], [1, 1, 2, 4, 4, 5, 5, 5, 8, 10], [1, 1, 2, 4, 4, 5, 5, 6, 8, 10], [1, 1, 2, 4, 5, 5, 5, 7, 8, 10], [1, 1, 2, 4, 6, 6, 7, 7, 8, 10], [1, 1, 2, 6, 6, 6, 7, 8, 8, 10], [1, 1, 3, 5, 5, 5, 5, 7, 9, 10], [1, 1, 3, 5, 6, 7, 8, 8, 8, 10], [1, 1, 4, 4, 5, 5, 8, 8, 8, 10], [1, 1, 4, 5, 5, 5, 7, 8, 9, 10], [1, 1, 4, 5, 5, 8, 8, 8, 8, 10], [1, 1, 4, 6, 7, 7, 8, 8, 9, 10], [1, 1, 4, 7, 8, 8, 8, 8, 8, 10], [1, 1, 5, 5, 5, 6, 6, 6, 10, 10], [1, 1, 6, 6, 6, 6, 7, 8, 10, 10], [1, 2, 2, 2, 3, 4, 4, 4, 5, 10], [1, 2, 2, 2, 3, 4, 4, 5, 6, 10], [1, 2, 2, 2, 3, 4, 6, 6, 6, 10], [1, 2, 2, 2, 4, 5, 5, 7, 7, 10], [1, 2, 2, 2, 4, 5, 6, 7, 7, 10], [1, 2, 2, 2, 5, 5, 7, 7, 7, 10], [1, 2, 2, 2, 6, 6, 6, 6, 8, 10], [1, 2, 2, 3, 4, 4, 5, 6, 8, 10], [1, 2, 2, 3, 4, 4, 6, 6, 8, 10], [1, 2, 2, 3, 4, 5, 6, 7, 8, 10], [1, 2, 2, 4, 4, 4, 4, 4, 8, 10], [1, 2, 2, 4, 4, 6, 6, 8, 8, 10], [1, 2, 2, 4, 5, 5, 5, 6, 9, 10], [1, 2, 2, 4, 5, 7, 7, 8, 8, 10], [1, 2, 2, 5, 5, 7, 7, 7, 9, 10], [1, 2, 2, 5, 7, 7, 8, 8, 8, 10], [1, 2, 2, 6, 6, 6, 7, 8, 9, 10], [1, 2, 3, 3, 3, 3, 3, 3, 5, 10], [1, 2, 3, 3, 4, 5, 5, 8, 8, 10], [1, 2, 3, 3, 5, 5, 6, 7, 9, 10], [1, 2, 3, 4, 4, 5, 7, 7, 9, 10], [1, 2, 3, 4, 4, 6, 8, 8, 8, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 6, 8, 8, 8, 8, 10], [1, 2, 3, 5, 6, 6, 6, 6, 10, 10], [1, 2, 3, 5, 7, 7, 8, 8, 9, 10], [1, 2, 3, 5, 8, 8, 8, 8, 8, 10], [1, 2, 4, 4, 4, 4, 4, 7, 9, 10], [1, 2, 4, 4, 4, 8, 8, 8, 8, 10], [1, 2, 4, 5, 5, 5, 5, 7, 10, 10], [1, 2, 4, 5, 5, 5, 7, 9, 9, 10], [1, 2, 4, 5, 6, 6, 8, 9, 9, 10], [1, 2, 5, 5, 5, 5, 7, 8, 10, 10], [1, 2, 5, 5, 7, 7, 8, 8, 10, 10], [1, 2, 5, 6, 6, 6, 9, 9, 9, 10], [1, 2, 5, 6, 6, 7, 9, 9, 9, 10], [1, 2, 5, 7, 7, 8, 8, 8, 10, 10], [1, 2, 6, 6, 6, 8, 9, 9, 9, 10], [1, 2, 6, 6, 8, 8, 8, 8, 10, 10], [1, 3, 3, 3, 3, 3, 3, 7, 7, 10], [1, 3, 3, 3, 3, 3, 5, 7, 8, 10], [1, 3, 3, 3, 3, 5, 5, 6, 9, 10], [1, 3, 3, 3, 3, 5, 7, 8, 8, 10], [1, 3, 3, 3, 4, 7, 8, 8, 8, 10], [1, 3, 3, 3, 5, 5, 7, 8, 9, 10], [1, 3, 3, 4, 4, 4, 6, 8, 9, 10], [1, 3, 3, 4, 6, 7, 7, 7, 10, 10], [1, 3, 3, 4, 6, 8, 8, 8, 9, 10], [1, 3, 3, 5, 5, 6, 6, 8, 10, 10], [1, 3, 3, 5, 7, 7, 7, 8, 10, 10], [1, 3, 3, 6, 6, 8, 8, 9, 9, 10], [1, 3, 3, 6, 7, 8, 8, 9, 9, 10], [1, 3, 4, 4, 4, 7, 7, 7, 10, 10], [1, 3, 4, 4, 4, 7, 7, 9, 9, 10], [1, 3, 4, 5, 6, 6, 7, 9, 10, 10], [1, 3, 4, 5, 6, 8, 8, 8, 10, 10], [1, 3, 4, 5, 7, 8, 8, 8, 10, 10], [1, 3, 4, 6, 8, 8, 8, 8, 10, 10], [1, 3, 5, 5, 5, 5, 7, 9, 10, 10], [1, 3, 5, 5, 8, 8, 9, 9, 9, 10], [1, 3, 5, 7, 8, 8, 8, 9, 10, 10], [1, 3, 6, 6, 7, 7, 7, 10, 10, 10], [1, 3, 6, 7, 7, 8, 9, 9, 10, 10], [1, 3, 6, 8, 8, 9, 9, 9, 9, 10], [1, 4, 4, 4, 4, 4, 4, 6, 10, 10], [1, 4, 4, 4, 5, 6, 9, 9, 9, 10], [1, 4, 4, 6, 7, 7, 9, 9, 10, 10], [1, 4, 4, 7, 8, 9, 9, 9, 9, 10], [1, 4, 5, 5, 5, 9, 9, 9, 9, 10], [1, 4, 5, 7, 7, 8, 8, 10, 10, 10], [1, 4, 5, 8, 8, 8, 9, 9, 10, 10], [1, 4, 6, 6, 6, 9, 9, 9, 10, 10], [1, 4, 6, 8, 8, 8, 8, 10, 10, 10], [1, 4, 8, 8, 8, 8, 8, 10, 10, 10], [1, 5, 5, 5, 5, 5, 7, 10, 10, 10], [1, 5, 5, 5, 5, 7, 8, 10, 10, 10], [1, 5, 5, 9, 9, 9, 9, 9, 9, 10], [1, 8, 8, 8, 8, 8, 10, 10, 10, 10], [1, 8, 8, 8, 9, 9, 9, 10, 10, 10]]
[[1, 1, 1, 2, 4, 5, 5, 6, 6, 6, 11], [1, 1, 1, 2, 5, 5, 5, 5, 6, 7, 11], [1, 1, 1, 3, 4, 5, 5, 7, 7, 7, 11], [1, 1, 1, 3, 5, 5, 6, 7, 7, 8, 11], [1, 1, 1, 3, 6, 6, 6, 6, 8, 8, 11], [1, 1, 1, 4, 4, 4, 4, 5, 7, 7, 11], [1, 1, 1, 4, 5, 5, 6, 6, 6, 9, 11], [1, 1, 1, 5, 5, 5, 5, 7, 7, 9, 11], [1, 1, 1, 5, 5, 6, 6, 7, 8, 9, 11], [1, 1, 2, 2, 3, 4, 6, 6, 6, 6, 11], [1, 1, 2, 2, 3, 5, 5, 6, 6, 7, 11], [1, 1, 2, 2, 4, 6, 6, 6, 7, 8, 11], [1, 1, 2, 2, 5, 5, 5, 5, 7, 8, 11], [1, 1, 2, 2, 6, 6, 6, 7, 8, 8, 11], [1, 1, 2, 3, 3, 4, 4, 5, 5, 7, 11], [1, 1, 2, 3, 3, 4, 6, 7, 7, 7, 11], [1, 1, 2, 3, 3, 6, 6, 7, 7, 8, 11], [1, 1, 2, 3, 4, 4, 5, 5, 7, 8, 11], [1, 1, 2, 3, 4, 6, 6, 6, 6, 9, 11], [1, 1, 2, 3, 5, 5, 6, 7, 7, 9, 11], [1, 1, 2, 3, 6, 6, 6, 7, 8, 9, 11], [1, 1, 2, 4, 4, 4, 5, 7, 8, 8, 11], [1, 1, 2, 4, 4, 5, 5, 5, 7, 9, 11], [1, 1, 2, 4, 5, 5, 5, 7, 8, 9, 11], [1, 1, 2, 4, 5, 7, 7, 8, 8, 9, 11], [1, 1, 2, 4, 7, 7, 8, 8, 8, 9, 11], [1, 1, 2, 5, 5, 5, 6, 6, 6, 10, 11], [1, 1, 2, 5, 6, 6, 6, 7, 8, 10, 11], [1, 1, 2, 5, 6, 6, 7, 7, 8, 10, 11], [1, 1, 2, 6, 6, 6, 7, 8, 8, 10, 11], [1, 1, 2, 7, 7, 7, 7, 9, 9, 9, 11], [1, 1, 3, 3, 3, 3, 3, 5, 6, 6, 11], [1, 1, 3, 3, 3, 3, 3, 6, 6, 6, 11], [1, 1, 3, 3, 3, 3, 4, 4, 4, 6, 11], [1, 1, 3, 3, 3, 3, 6, 6, 6, 8, 11], [1, 1, 3, 3, 3, 4, 4, 5, 5, 8, 11], [1, 1, 3, 3, 3, 5, 6, 6, 6, 9, 11], [1, 1, 3, 3, 3, 5, 7, 7, 8, 8, 11], [1, 1, 3, 3, 4, 5, 6, 6, 8, 9, 11], [1, 1, 3, 3, 4, 7, 7, 7, 8, 9, 11], [1, 1, 3, 3, 6, 6, 6, 7, 7, 10, 11], [1, 1, 3, 4, 4, 4, 4, 4, 8, 8, 11], [1, 1, 3, 4, 4, 4, 8, 8, 8, 8, 11], [1, 1, 3, 4, 4, 6, 6, 7, 9, 9, 11], [1, 1, 3, 4, 4, 6, 8, 8, 8, 9, 11], [1, 1, 3, 4, 4, 7, 8, 8, 8, 9, 11], [1, 1, 3, 4, 5, 5, 5, 7, 9, 9, 11], [1, 1, 3, 4, 6, 6, 6, 8, 8, 10, 11], [1, 1, 3, 4, 7, 8, 8, 8, 9, 9, 11], [1, 1, 3, 5, 5, 6, 7, 9, 9, 9, 11], [1, 1, 3, 5, 5, 7, 8, 8, 8, 10, 11], [1, 1, 3, 6, 6, 6, 8, 8, 9, 10, 11], [1, 1, 3, 6, 7, 8, 8, 8, 9, 10, 11], [1, 1, 4, 4, 4, 4, 6, 6, 7, 10, 11], [1, 1, 4, 4, 4, 5, 5, 5, 5, 10, 11], [1, 1, 4, 4, 4, 5, 6, 7, 8, 10, 11], [1, 1, 4, 4, 7, 7, 8, 8, 9, 10, 11], [1, 1, 4, 4, 8, 8, 8, 9, 9, 9, 11], [1, 1, 4, 5, 5, 5, 5, 7, 9, 10, 11], [1, 1, 4, 5, 5, 5, 7, 8, 9, 10, 11], [1, 1, 4, 5, 5, 6, 6, 6, 7, 11, 11], [1, 1, 4, 5, 5, 6, 6, 7, 7, 11, 11], [1, 1, 4, 5, 6, 6, 6, 7, 8, 11, 11], [1, 1, 4, 6, 6, 6, 7, 8, 10, 10, 11], [1, 1, 5, 5, 5, 5, 5, 6, 6, 11, 11], [1, 1, 5, 5, 5, 6, 6, 6, 10, 10, 11], [1, 1, 5, 5, 6, 6, 7, 7, 9, 11, 11], [1, 1, 5, 5, 6, 7, 9, 9, 9, 10, 11], [1, 1, 5, 5, 7, 7, 7, 8, 9, 11, 11], [1, 1, 5, 5, 7, 8, 9, 9, 9, 10, 11], [1, 1, 5, 9, 9, 9, 9, 9, 9, 9, 11], [1, 1, 6, 6, 6, 6, 7, 9, 9, 11, 11], [1, 1, 6, 6, 6, 7, 8, 9, 9, 11, 11], [1, 1, 7, 7, 8, 8, 9, 9, 9, 11, 11], [1, 1, 7, 8, 8, 8, 8, 10, 10, 10, 11], [1, 1, 8, 8, 8, 9, 9, 9, 10, 10, 11], [1, 2, 2, 2, 3, 3, 4, 5, 5, 5, 11], [1, 2, 2, 2, 3, 3, 6, 6, 6, 7, 11], [1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 11], [1, 2, 2, 2, 3, 4, 5, 7, 7, 7, 11], [1, 2, 2, 2, 3, 5, 5, 6, 7, 8, 11], [1, 2, 2, 2, 4, 4, 4, 5, 6, 8, 11], [1, 2, 2, 2, 4, 4, 7, 7, 7, 8, 11], [1, 2, 2, 2, 5, 5, 7, 7, 7, 9, 11], [1, 2, 2, 2, 7, 7, 7, 7, 8, 9, 11], [1, 2, 2, 3, 3, 3, 4, 4, 5, 7, 11], [1, 2, 2, 3, 3, 3, 4, 4, 6, 7, 11], [1, 2, 2, 3, 3, 3, 4, 5, 7, 7, 11], [1, 2, 2, 3, 3, 4, 4, 6, 7, 8, 11], [1, 2, 2, 3, 3, 6, 6, 7, 7, 9, 11], [1, 2, 2, 3, 4, 4, 5, 6, 7, 9, 11], [1, 2, 2, 3, 4, 5, 6, 7, 8, 9, 11], [1, 2, 2, 3, 5, 5, 7, 8, 8, 9, 11], [1, 2, 2, 3, 5, 6, 6, 6, 6, 10, 11], [1, 2, 2, 4, 4, 4, 5, 7, 8, 9, 11], [1, 2, 2, 4, 4, 5, 6, 6, 6, 10, 11], [1, 2, 2, 4, 4, 6, 7, 7, 7, 10, 11], [1, 2, 2, 5, 5, 7, 7, 7, 9, 10, 11], [1, 2, 2, 5, 6, 6, 7, 8, 9, 10, 11], [1, 2, 2, 5, 6, 8, 8, 9, 9, 9, 11], [1, 2, 2, 6, 6, 6, 7, 7, 7, 11, 11], [1, 2, 3, 3, 3, 3, 4, 4, 6, 8, 11], [1, 2, 3, 3, 3, 4, 4, 7, 8, 8, 11], [1, 2, 3, 3, 3, 4, 5, 7, 7, 9, 11], [1, 2, 3, 3, 3, 4, 7, 8, 8, 8, 11], [1, 2, 3, 3, 4, 5, 6, 7, 9, 9, 11], [1, 2, 3, 3, 6, 6, 7, 9, 9, 9, 11], [1, 2, 3, 3, 6, 7, 8, 8, 8, 10, 11], [1, 2, 3, 3, 8, 8, 8, 8, 9, 9, 11], [1, 2, 3, 4, 4, 4, 5, 5, 6, 10, 11], [1, 2, 3, 4, 4, 5, 5, 7, 8, 10, 11], [1, 2, 3, 4, 4, 8, 8, 8, 9, 9, 11], [1, 2, 3, 4, 5, 5, 6, 7, 9, 10, 11], [1, 2, 3, 4, 5, 5, 7, 7, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 6, 6, 6, 6, 7, 11, 11], [1, 2, 3, 4, 6, 6, 6, 7, 7, 11, 11], [1, 2, 3, 5, 5, 5, 5, 5, 9, 10, 11], [1, 2, 3, 5, 5, 5, 6, 6, 6, 11, 11], [1, 2, 3, 5, 5, 7, 7, 7, 8, 11, 11], [1, 2, 3, 5, 5, 7, 7, 9, 9, 10, 11], [1, 2, 3, 5, 6, 6, 6, 6, 10, 10, 11], [1, 2, 3, 6, 6, 6, 7, 7, 9, 11, 11], [1, 2, 3, 6, 6, 7, 9, 9, 9, 10, 11], [1, 2, 3, 6, 7, 7, 7, 8, 9, 11, 11], [1, 2, 3, 6, 7, 8, 9, 9, 9, 10, 11], [1, 2, 3, 7, 9, 9, 9, 9, 9, 9, 11], [1, 2, 4, 4, 4, 5, 5, 9, 9, 9, 11], [1, 2, 4, 4, 4, 5, 8, 9, 9, 9, 11], [1, 2, 4, 4, 6, 9, 9, 9, 9, 9, 11], [1, 2, 4, 4, 7, 7, 7, 7, 9, 11, 11], [1, 2, 4, 4, 8, 9, 9, 9, 9, 9, 11], [1, 2, 4, 5, 5, 6, 6, 7, 9, 11, 11], [1, 2, 4, 5, 5, 7, 9, 9, 9, 10, 11], [1, 2, 4, 5, 6, 7, 8, 9, 10, 10, 11], [1, 2, 4, 5, 7, 8, 8, 9, 10, 10, 11], [1, 2, 4, 6, 6, 6, 7, 9, 9, 11, 11], [1, 2, 4, 6, 7, 8, 8, 9, 9, 11, 11], [1, 2, 4, 7, 7, 7, 7, 10, 10, 10, 11], [1, 2, 4, 8, 8, 8, 8, 9, 9, 11, 11], [1, 2, 4, 8, 9, 9, 9, 9, 9, 10, 11], [1, 2, 5, 5, 6, 8, 9, 9, 10, 10, 11], [1, 2, 5, 6, 6, 6, 6, 10, 10, 10, 11], [1, 2, 5, 6, 6, 6, 8, 10, 10, 10, 11], [1, 2, 5, 6, 6, 9, 9, 9, 10, 10, 11], [1, 2, 5, 7, 8, 8, 9, 10, 10, 10, 11], [1, 2, 5, 7, 9, 9, 9, 9, 10, 10, 11], [1, 2, 6, 6, 6, 6, 8, 9, 10, 11, 11], [1, 2, 7, 7, 7, 8, 8, 10, 10, 11, 11], [1, 2, 7, 7, 8, 9, 9, 9, 10, 11, 11], [1, 2, 7, 8, 8, 9, 9, 9, 10, 11, 11], [1, 3, 3, 3, 3, 3, 3, 3, 3, 6, 11], [1, 3, 3, 3, 3, 3, 4, 5, 6, 9, 11], [1, 3, 3, 3, 3, 4, 4, 4, 7, 9, 11], [1, 3, 3, 3, 3, 4, 4, 6, 8, 9, 11], [1, 3, 3, 3, 3, 4, 7, 8, 8, 9, 11], [1, 3, 3, 3, 3, 5, 5, 6, 7, 10, 11], [1, 3, 3, 3, 3, 5, 5, 6, 9, 9, 11], [1, 3, 3, 3, 3, 6, 6, 6, 8, 10, 11], [1, 3, 3, 3, 3, 6, 7, 8, 9, 9, 11], [1, 3, 3, 3, 3, 7, 7, 8, 9, 9, 11], [1, 3, 3, 3, 4, 4, 5, 5, 5, 10, 11], [1, 3, 3, 3, 4, 4, 5, 5, 9, 9, 11], [1, 3, 3, 3, 4, 5, 5, 5, 8, 10, 11], [1, 3, 3, 3, 4, 5, 7, 8, 8, 10, 11], [1, 3, 3, 3, 4, 6, 6, 9, 9, 9, 11], [1, 3, 3, 3, 4, 7, 8, 8, 8, 10, 11], [1, 3, 3, 3, 6, 6, 9, 9, 9, 9, 11], [1, 3, 3, 3, 7, 7, 7, 9, 9, 10, 11], [1, 3, 3, 3, 7, 8, 9, 9, 9, 9, 11], [1, 3, 3, 4, 4, 5, 6, 8, 9, 10, 11], [1, 3, 3, 4, 5, 6, 7, 8, 8, 11, 11], [1, 3, 3, 4, 5, 6, 8, 9, 9, 10, 11], [1, 3, 3, 4, 7, 8, 8, 8, 10, 10, 11], [1, 3, 3, 5, 5, 5, 6, 8, 10, 10, 11], [1, 3, 3, 5, 6, 6, 8, 9, 10, 10, 11], [1, 3, 3, 5, 7, 7, 7, 9, 9, 11, 11], [1, 3, 3, 5, 8, 8, 8, 9, 10, 10, 11], [1, 3, 3, 6, 8, 8, 9, 9, 10, 10, 11], [1, 3, 4, 4, 4, 4, 6, 6, 7, 11, 11], [1, 3, 4, 4, 4, 5, 5, 5, 7, 11, 11], [1, 3, 4, 4, 4, 7, 8, 8, 8, 11, 11], [1, 3, 4, 4, 5, 5, 6, 7, 9, 11, 11], [1, 3, 4, 4, 5, 5, 8, 8, 10, 10, 11], [1, 3, 4, 4, 5, 6, 7, 9, 10, 10, 11], [1, 3, 4, 5, 6, 7, 9, 9, 9, 11, 11], [1, 3, 4, 5, 8, 8, 8, 10, 10, 10, 11], [1, 3, 4, 7, 7, 7, 9, 9, 10, 11, 11], [1, 3, 5, 5, 7, 7, 9, 9, 10, 11, 11], [1, 3, 5, 6, 6, 7, 7, 8, 11, 11, 11], [1, 3, 5, 6, 7, 8, 8, 8, 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5, 5, 6, 9, 10, 12], [1, 3, 3, 3, 3, 4, 6, 6, 6, 6, 11, 12], [1, 3, 3, 3, 3, 4, 8, 9, 9, 9, 9, 12], [1, 3, 3, 3, 3, 5, 5, 6, 7, 8, 11, 12], [1, 3, 3, 3, 3, 5, 5, 6, 9, 9, 10, 12], [1, 3, 3, 3, 3, 5, 7, 8, 8, 10, 10, 12], [1, 3, 3, 3, 3, 6, 6, 6, 6, 9, 11, 12], [1, 3, 3, 3, 3, 6, 6, 7, 8, 9, 11, 12], [1, 3, 3, 3, 3, 7, 8, 8, 8, 9, 11, 12], [1, 3, 3, 3, 4, 4, 5, 7, 8, 8, 11, 12], [1, 3, 3, 3, 4, 4, 6, 7, 7, 9, 11, 12], [1, 3, 3, 3, 4, 5, 5, 5, 8, 10, 10, 12], [1, 3, 3, 3, 4, 6, 7, 7, 10, 10, 10, 12], [1, 3, 3, 3, 4, 7, 8, 8, 8, 10, 11, 12], [1, 3, 3, 3, 4, 7, 8, 8, 10, 10, 10, 12], [1, 3, 3, 3, 5, 5, 7, 7, 7, 8, 12, 12], [1, 3, 3, 3, 5, 5, 8, 8, 10, 10, 10, 12], [1, 3, 3, 3, 6, 7, 7, 7, 9, 9, 12, 12], [1, 3, 3, 3, 6, 7, 8, 8, 10, 10, 11, 12], [1, 3, 3, 3, 6, 8, 9, 9, 9, 10, 11, 12], [1, 3, 3, 3, 7, 7, 9, 10, 10, 10, 10, 12], [1, 3, 3, 3, 8, 9, 9, 9, 9, 10, 11, 12], [1, 3, 3, 4, 4, 4, 5, 8, 8, 9, 11, 12], [1, 3, 3, 4, 4, 4, 7, 9, 9, 10, 10, 12], [1, 3, 3, 4, 4, 8, 8, 9, 9, 10, 11, 12], [1, 3, 3, 4, 5, 5, 5, 5, 7, 7, 12, 12], [1, 3, 3, 4, 5, 5, 5, 7, 9, 10, 11, 12], [1, 3, 3, 4, 5, 5, 6, 6, 7, 9, 12, 12], [1, 3, 3, 4, 5, 5, 6, 7, 8, 9, 12, 12], [1, 3, 3, 4, 5, 6, 6, 6, 7, 11, 11, 12], [1, 3, 3, 4, 5, 7, 7, 7, 8, 10, 12, 12], [1, 3, 3, 4, 5, 7, 7, 8, 9, 11, 11, 12], [1, 3, 3, 4, 5, 8, 8, 9, 10, 10, 11, 12], [1, 3, 3, 4, 6, 8, 8, 8, 9, 10, 12, 12], [1, 3, 3, 4, 8, 8, 8, 9, 9, 10, 12, 12], [1, 3, 3, 5, 5, 5, 6, 6, 9, 9, 12, 12], [1, 3, 3, 5, 5, 5, 7, 7, 8, 10, 12, 12], [1, 3, 3, 5, 5, 5, 7, 9, 10, 10, 11, 12], [1, 3, 3, 5, 5, 6, 6, 6, 6, 10, 12, 12], [1, 3, 3, 5, 5, 6, 7, 8, 9, 10, 12, 12], [1, 3, 3, 5, 5, 7, 7, 7, 10, 11, 11, 12], [1, 3, 3, 5, 5, 7, 9, 9, 9, 11, 11, 12], [1, 3, 3, 5, 6, 6, 6, 7, 10, 11, 11, 12], [1, 3, 3, 5, 6, 7, 7, 7, 8, 11, 12, 12], [1, 3, 3, 5, 7, 9, 9, 9, 10, 11, 11, 12], [1, 3, 3, 5, 8, 8, 8, 8, 9, 11, 12, 12], [1, 3, 3, 5, 9, 9, 9, 9, 9, 10, 12, 12], [1, 3, 3, 6, 6, 6, 6, 7, 8, 11, 12, 12], [1, 3, 3, 6, 6, 7, 9, 9, 10, 10, 12, 12], [1, 3, 3, 6, 6, 8, 8, 8, 11, 11, 11, 12], [1, 3, 3, 6, 7, 8, 9, 9, 9, 11, 12, 12], [1, 3, 3, 7, 7, 8, 8, 9, 10, 11, 12, 12], [1, 3, 3, 7, 7, 9, 9, 9, 11, 11, 11, 12], [1, 3, 3, 8, 8, 8, 9, 9, 10, 11, 12, 12], [1, 3, 3, 9, 9, 9, 9, 9, 9, 11, 12, 12], [1, 3, 4, 4, 4, 4, 4, 4, 5, 5, 11, 12], [1, 3, 4, 4, 4, 4, 4, 4, 6, 10, 10, 12], [1, 3, 4, 4, 4, 4, 5, 6, 6, 10, 11, 12], [1, 3, 4, 4, 4, 4, 6, 6, 6, 8, 12, 12], [1, 3, 4, 4, 4, 4, 7, 9, 10, 10, 10, 12], [1, 3, 4, 4, 4, 6, 7, 7, 9, 11, 11, 12], [1, 3, 4, 4, 4, 8, 8, 8, 8, 10, 12, 12], [1, 3, 4, 4, 5, 5, 5, 5, 7, 9, 12, 12], [1, 3, 4, 4, 5, 5, 5, 7, 8, 11, 11, 12], [1, 3, 4, 4, 5, 5, 6, 7, 9, 11, 11, 12], [1, 3, 4, 4, 5, 5, 8, 9, 9, 9, 12, 12], [1, 3, 4, 4, 5, 6, 6, 7, 9, 10, 12, 12], [1, 3, 4, 4, 5, 6, 6, 9, 9, 11, 11, 12], [1, 3, 4, 4, 5, 6, 10, 10, 10, 10, 10, 12], [1, 3, 4, 4, 6, 9, 9, 9, 9, 10, 12, 12], [1, 3, 4, 4, 6, 10, 10, 10, 10, 10, 10, 12], [1, 3, 4, 4, 7, 7, 8, 10, 10, 11, 11, 12], [1, 3, 4, 5, 5, 5, 5, 5, 10, 10, 11, 12], [1, 3, 4, 5, 5, 5, 5, 6, 8, 10, 12, 12], [1, 3, 4, 5, 5, 5, 6, 6, 9, 10, 12, 12], [1, 3, 4, 5, 5, 5, 7, 9, 9, 10, 12, 12], [1, 3, 4, 5, 5, 7, 7, 8, 9, 11, 12, 12], [1, 3, 4, 5, 5, 7, 8, 8, 9, 11, 12, 12], [1, 3, 4, 5, 5, 7, 10, 10, 10, 10, 11, 12], [1, 3, 4, 5, 6, 6, 6, 6, 6, 11, 12, 12], [1, 3, 4, 5, 6, 6, 6, 7, 9, 11, 12, 12], [1, 3, 4, 5, 6, 6, 7, 7, 11, 11, 11, 12], [1, 3, 4, 5, 6, 6, 7, 8, 11, 11, 11, 12], [1, 3, 4, 5, 6, 7, 7, 9, 11, 11, 11, 12], [1, 3, 4, 5, 6, 7, 8, 10, 10, 10, 12, 12], [1, 3, 4, 5, 6, 9, 9, 10, 10, 11, 11, 12], [1, 3, 4, 5, 7, 8, 8, 9, 10, 11, 12, 12], [1, 3, 4, 5, 9, 9, 9, 9, 9, 11, 12, 12], [1, 3, 4, 6, 6, 7, 8, 10, 11, 11, 11, 12], [1, 3, 4, 6, 6, 8, 9, 9, 10, 11, 12, 12], [1, 3, 4, 6, 7, 8, 8, 8, 9, 12, 12, 12], [1, 3, 4, 6, 7, 9, 10, 10, 10, 10, 12, 12], [1, 3, 4, 6, 9, 9, 9, 10, 11, 11, 11, 12], [1, 3, 4, 7, 7, 8, 8, 9, 11, 11, 12, 12], [1, 3, 4, 7, 9, 9, 9, 10, 10, 11, 12, 12], [1, 3, 4, 8, 8, 8, 8, 8, 10, 12, 12, 12], [1, 3, 4, 8, 8, 8, 9, 11, 11, 11, 11, 12], [1, 3, 4, 8, 8, 9, 9, 9, 9, 12, 12, 12], [1, 3, 4, 9, 9, 9, 10, 10, 11, 11, 11, 12], [1, 3, 5, 5, 5, 5, 7, 9, 10, 10, 12, 12], [1, 3, 5, 5, 5, 5, 8, 9, 10, 10, 12, 12], [1, 3, 5, 5, 5, 6, 6, 10, 10, 11, 11, 12], [1, 3, 5, 5, 5, 8, 9, 10, 10, 10, 12, 12], [1, 3, 5, 5, 6, 6, 6, 7, 10, 11, 12, 12], [1, 3, 5, 5, 6, 6, 6, 8, 10, 11, 12, 12], [1, 3, 5, 5, 6, 6, 7, 9, 10, 11, 12, 12], [1, 3, 5, 5, 6, 6, 10, 10, 10, 11, 11, 12], [1, 3, 5, 5, 7, 8, 9, 10, 10, 11, 12, 12], [1, 3, 5, 6, 6, 6, 6, 7, 8, 12, 12, 12], [1, 3, 5, 6, 6, 6, 6, 8, 8, 12, 12, 12], [1, 3, 5, 6, 6, 6, 7, 8, 9, 12, 12, 12], [1, 3, 5, 6, 6, 6, 8, 10, 10, 11, 12, 12], [1, 3, 5, 6, 6, 7, 10, 10, 11, 11, 11, 12], [1, 3, 5, 6, 7, 7, 9, 11, 11, 11, 11, 12], [1, 3, 5, 6, 7, 8, 8, 9, 10, 12, 12, 12], [1, 3, 5, 7, 8, 8, 8, 10, 10, 12, 12, 12], [1, 3, 5, 7, 9, 9, 9, 10, 11, 11, 12, 12], [1, 3, 5, 10, 10, 10, 10, 10, 10, 10, 12, 12], [1, 3, 6, 6, 6, 6, 6, 6, 6, 12, 12, 12], [1, 3, 6, 6, 6, 6, 6, 7, 11, 11, 12, 12], [1, 3, 6, 6, 6, 6, 8, 8, 10, 12, 12, 12], [1, 3, 6, 6, 6, 7, 8, 10, 11, 11, 12, 12], [1, 3, 6, 7, 8, 9, 9, 9, 11, 12, 12, 12], [1, 3, 6, 9, 9, 9, 9, 9, 11, 12, 12, 12], [1, 3, 6, 9, 9, 9, 9, 11, 11, 11, 12, 12], [1, 3, 6, 9, 10, 10, 10, 11, 11, 11, 11, 12], [1, 3, 7, 7, 8, 9, 10, 11, 11, 11, 12, 12], [1, 3, 7, 7, 8, 10, 11, 11, 11, 11, 11, 12], [1, 3, 8, 8, 9, 9, 10, 10, 11, 12, 12, 12], [1, 3, 8, 8, 10, 10, 10, 10, 10, 12, 12, 12], [1, 3, 9, 9, 9, 10, 10, 11, 11, 11, 12, 12], [1, 4, 4, 4, 4, 4, 4, 6, 6, 6, 12, 12], [1, 4, 4, 4, 4, 4, 4, 6, 10, 10, 10, 12], [1, 4, 4, 4, 4, 4, 4, 8, 10, 10, 10, 12], [1, 4, 4, 4, 4, 4, 8, 10, 10, 10, 10, 12], [1, 4, 4, 4, 4, 5, 5, 5, 8, 9, 12, 12], [1, 4, 4, 4, 4, 5, 5, 6, 6, 11, 11, 12], [1, 4, 4, 4, 4, 5, 5, 6, 8, 11, 11, 12], [1, 4, 4, 4, 4, 5, 6, 9, 9, 9, 12, 12], [1, 4, 4, 4, 5, 5, 6, 8, 10, 11, 11, 12], [1, 4, 4, 4, 5, 6, 6, 8, 8, 11, 12, 12], [1, 4, 4, 4, 5, 8, 9, 9, 10, 10, 12, 12], [1, 4, 4, 4, 6, 6, 9, 10, 10, 11, 11, 12], [1, 4, 4, 4, 6, 7, 7, 7, 10, 11, 12, 12], [1, 4, 4, 4, 6, 8, 9, 9, 11, 11, 11, 12], [1, 4, 4, 4, 7, 8, 8, 10, 11, 11, 11, 12], [1, 4, 4, 4, 9, 9, 9, 10, 10, 10, 12, 12], [1, 4, 4, 5, 5, 6, 6, 9, 9, 11, 12, 12], [1, 4, 4, 5, 5, 6, 9, 9, 9, 11, 12, 12], [1, 4, 4, 5, 5, 8, 10, 10, 10, 11, 11, 12], [1, 4, 4, 5, 6, 8, 8, 10, 10, 11, 12, 12], [1, 4, 4, 5, 6, 9, 9, 10, 11, 11, 11, 12], [1, 4, 4, 5, 7, 7, 7, 7, 11, 11, 12, 12], [1, 4, 4, 5, 8, 8, 9, 9, 9, 12, 12, 12], [1, 4, 4, 5, 8, 9, 10, 10, 11, 11, 11, 12], [1, 4, 4, 6, 6, 8, 8, 8, 10, 12, 12, 12], [1, 4, 4, 6, 6, 8, 9, 9, 11, 11, 12, 12], [1, 4, 4, 6, 7, 10, 10, 10, 11, 11, 11, 12], [1, 4, 4, 6, 8, 8, 9, 10, 11, 11, 12, 12], [1, 4, 4, 6, 10, 10, 10, 10, 10, 10, 12, 12], [1, 4, 4, 7, 8, 8, 9, 10, 10, 12, 12, 12], [1, 4, 4, 8, 8, 9, 10, 10, 11, 11, 12, 12], [1, 4, 5, 5, 5, 5, 9, 9, 11, 11, 11, 12], [1, 4, 5, 5, 6, 7, 7, 7, 10, 12, 12, 12], [1, 4, 5, 5, 6, 9, 9, 9, 11, 11, 12, 12], [1, 4, 5, 6, 6, 10, 10, 10, 10, 11, 12, 12], [1, 4, 5, 6, 7, 10, 10, 11, 11, 11, 11, 12], [1, 4, 5, 7, 8, 8, 9, 10, 11, 12, 12, 12], [1, 4, 5, 8, 9, 9, 9, 10, 11, 12, 12, 12], [1, 4, 5, 8, 9, 9, 10, 11, 11, 11, 12, 12], [1, 4, 6, 6, 6, 8, 10, 10, 10, 12, 12, 12], [1, 4, 6, 6, 7, 8, 10, 11, 11, 11, 12, 12], [1, 4, 6, 6, 9, 9, 10, 11, 11, 11, 12, 12], [1, 4, 6, 6, 9, 10, 11, 11, 11, 11, 11, 12], [1, 4, 6, 7, 8, 8, 8, 9, 12, 12, 12, 12], [1, 4, 6, 8, 8, 9, 9, 9, 12, 12, 12, 12], [1, 4, 6, 8, 8, 9, 9, 11, 11, 12, 12, 12], [1, 4, 6, 8, 9, 11, 11, 11, 11, 11, 11, 12], [1, 4, 6, 10, 10, 10, 11, 11, 11, 11, 11, 12], [1, 4, 7, 7, 7, 10, 10, 10, 11, 12, 12, 12], [1, 4, 7, 9, 9, 9, 9, 10, 12, 12, 12, 12], [1, 4, 8, 8, 8, 8, 10, 10, 12, 12, 12, 12], [1, 4, 8, 8, 9, 10, 10, 11, 11, 12, 12, 12], [1, 4, 8, 9, 9, 10, 10, 11, 11, 12, 12, 12], [1, 5, 5, 5, 5, 5, 7, 7, 9, 12, 12, 12], [1, 5, 5, 5, 5, 6, 8, 11, 11, 11, 11, 12], [1, 5, 5, 5, 5, 7, 7, 9, 10, 12, 12, 12], [1, 5, 5, 5, 7, 7, 7, 9, 11, 12, 12, 12], [1, 5, 5, 5, 7, 8, 9, 11, 11, 11, 12, 12], [1, 5, 5, 5, 8, 9, 10, 10, 10, 12, 12, 12], [1, 5, 5, 5, 8, 9, 11, 11, 11, 11, 11, 12], [1, 5, 5, 6, 6, 6, 7, 7, 11, 12, 12, 12], [1, 5, 5, 6, 6, 7, 11, 11, 11, 11, 11, 12], [1, 5, 5, 6, 6, 10, 10, 10, 11, 11, 12, 12], [1, 5, 5, 7, 7, 7, 8, 8, 12, 12, 12, 12], [1, 5, 6, 6, 6, 8, 10, 10, 11, 12, 12, 12], [1, 5, 6, 6, 7, 9, 9, 9, 12, 12, 12, 12], [1, 5, 7, 7, 8, 9, 9, 11, 12, 12, 12, 12], [1, 5, 7, 7, 8, 10, 10, 10, 12, 12, 12, 12], [1, 5, 8, 9, 10, 10, 11, 11, 11, 12, 12, 12], [1, 6, 6, 6, 6, 8, 8, 10, 12, 12, 12, 12], [1, 6, 6, 7, 7, 9, 11, 11, 11, 12, 12, 12], [1, 6, 6, 10, 10, 11, 11, 11, 11, 11, 12, 12], [1, 6, 7, 9, 9, 9, 11, 11, 12, 12, 12, 12], [1, 6, 8, 8, 9, 10, 11, 11, 12, 12, 12, 12], [1, 6, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12], [1, 7, 7, 7, 7, 11, 11, 11, 11, 12, 12, 12], [1, 7, 8, 9, 10, 10, 10, 12, 12, 12, 12, 12], [1, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12]]
において、
1≦s2≦s3≦s4≦･･･≦s(n-1)≦n
なる自然数の組[1,s2,s3,s4,･･･,s(n-1),n]
において関係式
(1+s2+s3+s4+･･･+s(n-1)+n)^2=1^3+s2^3+s3^3+s4^3+･･･+s(n-1)^3+n^3
が成立する組合せ
を発見して欲しいとあり、n が2以上12以下の場合についてすべて求めてみた。
(2..12).each{|n|
a_ary = []
(1..n).to_a.repeated_combination(n){|c|
# 最初が1、最後がn
if c[0] == 1 && c[-1] == n
a_ary << c if c.inject(:+) ** 2 == c.map{|i| i * i * i}.inject(:+)
end
}
p a_ary
}
出力結果
[[1, 2]]
[[1, 2, 3]]
[[1, 2, 2, 4], [1, 2, 3, 4]]
[[1, 2, 2, 3, 5], [1, 2, 3, 4, 5]]
[[1, 2, 2, 3, 4, 6], [1, 2, 2, 4, 4, 6], [1, 2, 3, 4, 5, 6], [1, 4, 4, 4, 6, 6]]
[[1, 1, 3, 4, 4, 5, 7], [1, 1, 4, 5, 5, 5, 7], [1, 2, 2, 3, 3, 3, 7], [1, 2, 2, 3, 4, 5, 7], [1, 2, 3, 4, 5, 6, 7], [1, 3, 5, 5, 5, 7, 7], [1, 4, 4, 5, 6, 7, 7], [1, 5, 5, 6, 6, 7, 7]]
[[1, 1, 2, 4, 4, 5, 5, 8], [1, 1, 2, 4, 5, 5, 5, 8], [1, 1, 3, 3, 3, 4, 5, 8], [1, 1, 4, 5, 5, 5, 7, 8], [1, 2, 2, 2, 4, 4, 4, 8], [1, 2, 2, 3, 4, 4, 6, 8], [1, 2, 2, 3, 4, 5, 6, 8], [1, 2, 2, 4, 4, 6, 6, 8], [1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 4, 5, 5, 7, 7, 8], [1, 2, 5, 6, 6, 7, 7, 8], [1, 3, 3, 3, 3, 5, 7, 8], [1, 3, 3, 3, 6, 6, 7, 8], [1, 3, 4, 4, 4, 5, 8, 8], [1, 3, 4, 4, 5, 6, 8, 8], [1, 3, 4, 6, 6, 6, 8, 8], [1, 3, 4, 6, 7, 7, 7, 8], [1, 4, 5, 5, 7, 7, 8, 8], [1, 4, 5, 6, 7, 7, 8, 8], [1, 5, 5, 7, 7, 7, 8, 8], [1, 6, 6, 6, 6, 8, 8, 8]]
[[1, 1, 1, 4, 5, 5, 5, 5, 9], [1, 1, 2, 3, 4, 5, 5, 6, 9], [1, 1, 2, 4, 5, 5, 5, 7, 9], [1, 1, 2, 5, 5, 6, 6, 7, 9], [1, 1, 3, 3, 3, 3, 5, 5, 9], [1, 1, 3, 3, 3, 4, 4, 6, 9], [1, 1, 3, 3, 3, 5, 6, 6, 9], [1, 1, 3, 4, 5, 6, 7, 7, 9], [1, 1, 4, 5, 5, 5, 7, 8, 9], [1, 1, 4, 5, 5, 6, 7, 8, 9], [1, 1, 5, 5, 5, 7, 7, 8, 9], [1, 1, 6, 6, 7, 7, 7, 8, 9], [1, 2, 2, 2, 3, 5, 5, 5, 9], [1, 2, 2, 3, 3, 4, 6, 6, 9], [1, 2, 2, 3, 4, 4, 5, 7, 9], [1, 2, 2, 3, 4, 5, 6, 7, 9], [1, 2, 2, 3, 6, 6, 6, 7, 9], [1, 2, 2, 5, 5, 5, 6, 8, 9], [1, 2, 2, 5, 5, 7, 7, 7, 9], [1, 2, 2, 5, 6, 7, 7, 7, 9], [1, 2, 3, 3, 3, 3, 3, 3, 9], [1, 2, 3, 3, 3, 3, 3, 6, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 6, 6, 7, 8, 9], [1, 2, 3, 5, 6, 7, 7, 8, 9], [1, 2, 4, 4, 4, 4, 7, 8, 9], [1, 2, 4, 5, 6, 6, 6, 9, 9], [1, 2, 4, 6, 6, 7, 8, 8, 9], [1, 2, 5, 5, 5, 6, 7, 9, 9], [1, 2, 5, 7, 7, 7, 8, 8, 9], [1, 3, 3, 3, 3, 5, 5, 8, 9], [1, 3, 3, 4, 6, 6, 8, 8, 9], [1, 3, 3, 5, 5, 7, 8, 8, 9], [1, 3, 4, 5, 7, 7, 7, 9, 9], [1, 3, 4, 6, 7, 8, 8, 8, 9], [1, 3, 5, 6, 7, 7, 8, 9, 9], [1, 3, 6, 7, 8, 8, 8, 8, 9], [1, 4, 4, 4, 4, 7, 7, 9, 9], [1, 4, 4, 7, 8, 8, 8, 8, 9], [1, 4, 5, 6, 6, 6, 9, 9, 9], [1, 5, 5, 5, 7, 7, 9, 9, 9], [1, 5, 6, 6, 7, 8, 9, 9, 9]]
[[1, 1, 1, 4, 4, 4, 4, 5, 6, 10], [1, 1, 1, 4, 4, 5, 5, 5, 7, 10], [1, 1, 1, 4, 6, 6, 6, 7, 7, 10], [1, 1, 2, 2, 4, 4, 5, 5, 5, 10], [1, 1, 2, 2, 4, 5, 5, 6, 6, 10], [1, 1, 2, 3, 4, 4, 5, 6, 7, 10], [1, 1, 2, 4, 4, 5, 5, 5, 8, 10], [1, 1, 2, 4, 4, 5, 5, 6, 8, 10], [1, 1, 2, 4, 5, 5, 5, 7, 8, 10], [1, 1, 2, 4, 6, 6, 7, 7, 8, 10], [1, 1, 2, 6, 6, 6, 7, 8, 8, 10], [1, 1, 3, 5, 5, 5, 5, 7, 9, 10], [1, 1, 3, 5, 6, 7, 8, 8, 8, 10], [1, 1, 4, 4, 5, 5, 8, 8, 8, 10], [1, 1, 4, 5, 5, 5, 7, 8, 9, 10], [1, 1, 4, 5, 5, 8, 8, 8, 8, 10], [1, 1, 4, 6, 7, 7, 8, 8, 9, 10], [1, 1, 4, 7, 8, 8, 8, 8, 8, 10], [1, 1, 5, 5, 5, 6, 6, 6, 10, 10], [1, 1, 6, 6, 6, 6, 7, 8, 10, 10], [1, 2, 2, 2, 3, 4, 4, 4, 5, 10], [1, 2, 2, 2, 3, 4, 4, 5, 6, 10], [1, 2, 2, 2, 3, 4, 6, 6, 6, 10], [1, 2, 2, 2, 4, 5, 5, 7, 7, 10], [1, 2, 2, 2, 4, 5, 6, 7, 7, 10], [1, 2, 2, 2, 5, 5, 7, 7, 7, 10], [1, 2, 2, 2, 6, 6, 6, 6, 8, 10], [1, 2, 2, 3, 4, 4, 5, 6, 8, 10], [1, 2, 2, 3, 4, 4, 6, 6, 8, 10], [1, 2, 2, 3, 4, 5, 6, 7, 8, 10], [1, 2, 2, 4, 4, 4, 4, 4, 8, 10], [1, 2, 2, 4, 4, 6, 6, 8, 8, 10], [1, 2, 2, 4, 5, 5, 5, 6, 9, 10], [1, 2, 2, 4, 5, 7, 7, 8, 8, 10], [1, 2, 2, 5, 5, 7, 7, 7, 9, 10], [1, 2, 2, 5, 7, 7, 8, 8, 8, 10], [1, 2, 2, 6, 6, 6, 7, 8, 9, 10], [1, 2, 3, 3, 3, 3, 3, 3, 5, 10], [1, 2, 3, 3, 4, 5, 5, 8, 8, 10], [1, 2, 3, 3, 5, 5, 6, 7, 9, 10], [1, 2, 3, 4, 4, 5, 7, 7, 9, 10], [1, 2, 3, 4, 4, 6, 8, 8, 8, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 6, 8, 8, 8, 8, 10], [1, 2, 3, 5, 6, 6, 6, 6, 10, 10], [1, 2, 3, 5, 7, 7, 8, 8, 9, 10], [1, 2, 3, 5, 8, 8, 8, 8, 8, 10], [1, 2, 4, 4, 4, 4, 4, 7, 9, 10], [1, 2, 4, 4, 4, 8, 8, 8, 8, 10], [1, 2, 4, 5, 5, 5, 5, 7, 10, 10], [1, 2, 4, 5, 5, 5, 7, 9, 9, 10], [1, 2, 4, 5, 6, 6, 8, 9, 9, 10], [1, 2, 5, 5, 5, 5, 7, 8, 10, 10], [1, 2, 5, 5, 7, 7, 8, 8, 10, 10], [1, 2, 5, 6, 6, 6, 9, 9, 9, 10], [1, 2, 5, 6, 6, 7, 9, 9, 9, 10], [1, 2, 5, 7, 7, 8, 8, 8, 10, 10], [1, 2, 6, 6, 6, 8, 9, 9, 9, 10], [1, 2, 6, 6, 8, 8, 8, 8, 10, 10], [1, 3, 3, 3, 3, 3, 3, 7, 7, 10], [1, 3, 3, 3, 3, 3, 5, 7, 8, 10], [1, 3, 3, 3, 3, 5, 5, 6, 9, 10], [1, 3, 3, 3, 3, 5, 7, 8, 8, 10], [1, 3, 3, 3, 4, 7, 8, 8, 8, 10], [1, 3, 3, 3, 5, 5, 7, 8, 9, 10], [1, 3, 3, 4, 4, 4, 6, 8, 9, 10], [1, 3, 3, 4, 6, 7, 7, 7, 10, 10], [1, 3, 3, 4, 6, 8, 8, 8, 9, 10], [1, 3, 3, 5, 5, 6, 6, 8, 10, 10], [1, 3, 3, 5, 7, 7, 7, 8, 10, 10], [1, 3, 3, 6, 6, 8, 8, 9, 9, 10], [1, 3, 3, 6, 7, 8, 8, 9, 9, 10], [1, 3, 4, 4, 4, 7, 7, 7, 10, 10], [1, 3, 4, 4, 4, 7, 7, 9, 9, 10], [1, 3, 4, 5, 6, 6, 7, 9, 10, 10], [1, 3, 4, 5, 6, 8, 8, 8, 10, 10], [1, 3, 4, 5, 7, 8, 8, 8, 10, 10], [1, 3, 4, 6, 8, 8, 8, 8, 10, 10], [1, 3, 5, 5, 5, 5, 7, 9, 10, 10], [1, 3, 5, 5, 8, 8, 9, 9, 9, 10], [1, 3, 5, 7, 8, 8, 8, 9, 10, 10], [1, 3, 6, 6, 7, 7, 7, 10, 10, 10], [1, 3, 6, 7, 7, 8, 9, 9, 10, 10], [1, 3, 6, 8, 8, 9, 9, 9, 9, 10], [1, 4, 4, 4, 4, 4, 4, 6, 10, 10], [1, 4, 4, 4, 5, 6, 9, 9, 9, 10], [1, 4, 4, 6, 7, 7, 9, 9, 10, 10], [1, 4, 4, 7, 8, 9, 9, 9, 9, 10], [1, 4, 5, 5, 5, 9, 9, 9, 9, 10], [1, 4, 5, 7, 7, 8, 8, 10, 10, 10], [1, 4, 5, 8, 8, 8, 9, 9, 10, 10], [1, 4, 6, 6, 6, 9, 9, 9, 10, 10], [1, 4, 6, 8, 8, 8, 8, 10, 10, 10], [1, 4, 8, 8, 8, 8, 8, 10, 10, 10], [1, 5, 5, 5, 5, 5, 7, 10, 10, 10], [1, 5, 5, 5, 5, 7, 8, 10, 10, 10], [1, 5, 5, 9, 9, 9, 9, 9, 9, 10], [1, 8, 8, 8, 8, 8, 10, 10, 10, 10], [1, 8, 8, 8, 9, 9, 9, 10, 10, 10]]
[[1, 1, 1, 2, 4, 5, 5, 6, 6, 6, 11], [1, 1, 1, 2, 5, 5, 5, 5, 6, 7, 11], [1, 1, 1, 3, 4, 5, 5, 7, 7, 7, 11], [1, 1, 1, 3, 5, 5, 6, 7, 7, 8, 11], [1, 1, 1, 3, 6, 6, 6, 6, 8, 8, 11], [1, 1, 1, 4, 4, 4, 4, 5, 7, 7, 11], [1, 1, 1, 4, 5, 5, 6, 6, 6, 9, 11], [1, 1, 1, 5, 5, 5, 5, 7, 7, 9, 11], [1, 1, 1, 5, 5, 6, 6, 7, 8, 9, 11], [1, 1, 2, 2, 3, 4, 6, 6, 6, 6, 11], [1, 1, 2, 2, 3, 5, 5, 6, 6, 7, 11], [1, 1, 2, 2, 4, 6, 6, 6, 7, 8, 11], [1, 1, 2, 2, 5, 5, 5, 5, 7, 8, 11], [1, 1, 2, 2, 6, 6, 6, 7, 8, 8, 11], [1, 1, 2, 3, 3, 4, 4, 5, 5, 7, 11], [1, 1, 2, 3, 3, 4, 6, 7, 7, 7, 11], [1, 1, 2, 3, 3, 6, 6, 7, 7, 8, 11], [1, 1, 2, 3, 4, 4, 5, 5, 7, 8, 11], [1, 1, 2, 3, 4, 6, 6, 6, 6, 9, 11], [1, 1, 2, 3, 5, 5, 6, 7, 7, 9, 11], [1, 1, 2, 3, 6, 6, 6, 7, 8, 9, 11], [1, 1, 2, 4, 4, 4, 5, 7, 8, 8, 11], [1, 1, 2, 4, 4, 5, 5, 5, 7, 9, 11], [1, 1, 2, 4, 5, 5, 5, 7, 8, 9, 11], [1, 1, 2, 4, 5, 7, 7, 8, 8, 9, 11], [1, 1, 2, 4, 7, 7, 8, 8, 8, 9, 11], [1, 1, 2, 5, 5, 5, 6, 6, 6, 10, 11], [1, 1, 2, 5, 6, 6, 6, 7, 8, 10, 11], [1, 1, 2, 5, 6, 6, 7, 7, 8, 10, 11], [1, 1, 2, 6, 6, 6, 7, 8, 8, 10, 11], [1, 1, 2, 7, 7, 7, 7, 9, 9, 9, 11], [1, 1, 3, 3, 3, 3, 3, 5, 6, 6, 11], [1, 1, 3, 3, 3, 3, 3, 6, 6, 6, 11], [1, 1, 3, 3, 3, 3, 4, 4, 4, 6, 11], [1, 1, 3, 3, 3, 3, 6, 6, 6, 8, 11], [1, 1, 3, 3, 3, 4, 4, 5, 5, 8, 11], [1, 1, 3, 3, 3, 5, 6, 6, 6, 9, 11], [1, 1, 3, 3, 3, 5, 7, 7, 8, 8, 11], [1, 1, 3, 3, 4, 5, 6, 6, 8, 9, 11], [1, 1, 3, 3, 4, 7, 7, 7, 8, 9, 11], [1, 1, 3, 3, 6, 6, 6, 7, 7, 10, 11], [1, 1, 3, 4, 4, 4, 4, 4, 8, 8, 11], [1, 1, 3, 4, 4, 4, 8, 8, 8, 8, 11], [1, 1, 3, 4, 4, 6, 6, 7, 9, 9, 11], [1, 1, 3, 4, 4, 6, 8, 8, 8, 9, 11], [1, 1, 3, 4, 4, 7, 8, 8, 8, 9, 11], [1, 1, 3, 4, 5, 5, 5, 7, 9, 9, 11], [1, 1, 3, 4, 6, 6, 6, 8, 8, 10, 11], [1, 1, 3, 4, 7, 8, 8, 8, 9, 9, 11], [1, 1, 3, 5, 5, 6, 7, 9, 9, 9, 11], [1, 1, 3, 5, 5, 7, 8, 8, 8, 10, 11], [1, 1, 3, 6, 6, 6, 8, 8, 9, 10, 11], [1, 1, 3, 6, 7, 8, 8, 8, 9, 10, 11], [1, 1, 4, 4, 4, 4, 6, 6, 7, 10, 11], [1, 1, 4, 4, 4, 5, 5, 5, 5, 10, 11], [1, 1, 4, 4, 4, 5, 6, 7, 8, 10, 11], [1, 1, 4, 4, 7, 7, 8, 8, 9, 10, 11], [1, 1, 4, 4, 8, 8, 8, 9, 9, 9, 11], [1, 1, 4, 5, 5, 5, 5, 7, 9, 10, 11], [1, 1, 4, 5, 5, 5, 7, 8, 9, 10, 11], [1, 1, 4, 5, 5, 6, 6, 6, 7, 11, 11], [1, 1, 4, 5, 5, 6, 6, 7, 7, 11, 11], [1, 1, 4, 5, 6, 6, 6, 7, 8, 11, 11], [1, 1, 4, 6, 6, 6, 7, 8, 10, 10, 11], [1, 1, 5, 5, 5, 5, 5, 6, 6, 11, 11], [1, 1, 5, 5, 5, 6, 6, 6, 10, 10, 11], [1, 1, 5, 5, 6, 6, 7, 7, 9, 11, 11], [1, 1, 5, 5, 6, 7, 9, 9, 9, 10, 11], [1, 1, 5, 5, 7, 7, 7, 8, 9, 11, 11], [1, 1, 5, 5, 7, 8, 9, 9, 9, 10, 11], [1, 1, 5, 9, 9, 9, 9, 9, 9, 9, 11], [1, 1, 6, 6, 6, 6, 7, 9, 9, 11, 11], [1, 1, 6, 6, 6, 7, 8, 9, 9, 11, 11], [1, 1, 7, 7, 8, 8, 9, 9, 9, 11, 11], [1, 1, 7, 8, 8, 8, 8, 10, 10, 10, 11], [1, 1, 8, 8, 8, 9, 9, 9, 10, 10, 11], [1, 2, 2, 2, 3, 3, 4, 5, 5, 5, 11], [1, 2, 2, 2, 3, 3, 6, 6, 6, 7, 11], [1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 11], [1, 2, 2, 2, 3, 4, 5, 7, 7, 7, 11], [1, 2, 2, 2, 3, 5, 5, 6, 7, 8, 11], [1, 2, 2, 2, 4, 4, 4, 5, 6, 8, 11], [1, 2, 2, 2, 4, 4, 7, 7, 7, 8, 11], [1, 2, 2, 2, 5, 5, 7, 7, 7, 9, 11], [1, 2, 2, 2, 7, 7, 7, 7, 8, 9, 11], [1, 2, 2, 3, 3, 3, 4, 4, 5, 7, 11], [1, 2, 2, 3, 3, 3, 4, 4, 6, 7, 11], [1, 2, 2, 3, 3, 3, 4, 5, 7, 7, 11], [1, 2, 2, 3, 3, 4, 4, 6, 7, 8, 11], [1, 2, 2, 3, 3, 6, 6, 7, 7, 9, 11], [1, 2, 2, 3, 4, 4, 5, 6, 7, 9, 11], [1, 2, 2, 3, 4, 5, 6, 7, 8, 9, 11], [1, 2, 2, 3, 5, 5, 7, 8, 8, 9, 11], [1, 2, 2, 3, 5, 6, 6, 6, 6, 10, 11], [1, 2, 2, 4, 4, 4, 5, 7, 8, 9, 11], [1, 2, 2, 4, 4, 5, 6, 6, 6, 10, 11], [1, 2, 2, 4, 4, 6, 7, 7, 7, 10, 11], [1, 2, 2, 5, 5, 7, 7, 7, 9, 10, 11], [1, 2, 2, 5, 6, 6, 7, 8, 9, 10, 11], [1, 2, 2, 5, 6, 8, 8, 9, 9, 9, 11], [1, 2, 2, 6, 6, 6, 7, 7, 7, 11, 11], [1, 2, 3, 3, 3, 3, 4, 4, 6, 8, 11], [1, 2, 3, 3, 3, 4, 4, 7, 8, 8, 11], [1, 2, 3, 3, 3, 4, 5, 7, 7, 9, 11], [1, 2, 3, 3, 3, 4, 7, 8, 8, 8, 11], [1, 2, 3, 3, 4, 5, 6, 7, 9, 9, 11], [1, 2, 3, 3, 6, 6, 7, 9, 9, 9, 11], [1, 2, 3, 3, 6, 7, 8, 8, 8, 10, 11], [1, 2, 3, 3, 8, 8, 8, 8, 9, 9, 11], [1, 2, 3, 4, 4, 4, 5, 5, 6, 10, 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