求所有实数 α ，使得对任何正整数 n ，[α]+[2α]+…+[nα] 均为 n 的倍数

cgl_74

看看有哪些解法

ccmmjj126

α是个偶数就可以了。

uk702

探索思路：假设 α = k + β，其中 β 为小数且不等于 0，将 β 表示成二进制，对于给定的 β，是否总可以找到一个 m，使得 n = 2^m 时， [α] + [2α] + ... + [nα]  不是 n = 2^m 的倍数？

比如说，若 β >= 1/2，要使得 [α] + [2α]  是 2 的倍数，则 k 必须是奇数。

cgl_74

求所有解，还有其它解呢？

王守恩

\(求所有实数\ α\ 满足:\ 对任意正整数\ n,\ 整数\ \lfloorα\rfloor+\lfloor2α\rfloor+…+\lfloor nα\rfloor\ 均为\ n\ 的倍数。\lfloor A\rfloor\ 表示不超过\ A\ 的最大整数。\)

\(1,\ 答案。\ α=\{...-8,-6,-4,-2,0,2,4,6,8,...\}。\ α\ 不能是奇数。\ α\ 不能是小数。\)

1. Table[Mod[Sum[2a k, {k, n}], n], {a, -4, 4}, {n, 1, 9}]

复制代码

{{0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}}

1. Table[Mod[a n (n + 1), n], {a, -4, 4}, {n, 1, 9}]

复制代码

{{0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}}

\(2,\ 假设\  α = \ 整数部分 + β(小数部分)，\ 其中\ 0< β<1。\ 记\ S(β,n)=\lfloorβ\rfloor+\lfloor2β\rfloor+…+\lfloor nβ\rfloor。则\)

     \(S(β,1)\ 肯定是\ 1\ 的倍数。S(β,2)\ 肯定是\ 2\ 的倍数。S(β,3)\ 肯定是\ 3\ 的倍数。S(β,4)\ 肯定是\ 4\ 的倍数。...\)

    \(S(β,1):没问题。S(β,2):β<1/2\ 是不行的。S(β,3):β<2/3\ 是不行的。S(β,4):β<3/4\ 是不行的。S(β,5):β<4/5\ 是不行的。...\)

\(3,\ 对模\  n\  来说,\ 2 p + β ≡ \ 2 p - β ≡ β,\ \ 这里取β = a/b,\ b≥2, \ 1≤a≤b-1,\ \ 当然, \ \ β\ 可以取无理数。\)

1. Table[Mod[Sum[Floor[k (a/b)], {k, n}], n], {b, 2, 19}, {a, b - 1}, {n, b}]

复制代码

b=2,{{0, 1}},
b=3,{{0, 0, 1}, {0, 1, 0}},
b=4,{{0, 0, 0, 1}, {0, 1, 2, 0}, {0, 1, 0, 2}},
b=5,{{0, 0, 0, 0, 1}, {0, 0, 1, 2, 4}, {0, 1, 2, 0, 2}, {0, 1,0, 2, 0}},
b=6,{{0, 0, 0, 0, 0, 1}, {0, 0, 1, 2, 3, 5}, {0, 1, 2, 0, 1, 3}, {0, 1, 0, 1, 3, 0}, {0, 1, 0, 2, 0, 3}},
b=7,{{0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 1, 2, 3, 5}, {0, 0, 1, 2, 4, 0, 2}, {0, 1, 2, 0, 1, 3, 6}, {0, 1, 0, 1, 3, 0, 3}, {0, 1, 0, 2, 0, 3, 0}},
b=8,{{0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 1, 2, 3, 4, 6}, {0, 0, 1, 2, 3, 5, 0, 2}, {0, 1, 2, 0, 1, 3, 5, 0}, {0, 1, 2, 0, 2, 4, 0, 3}, {0, 1, 0, 2, 4, 1, 4, 0}, {0, 1, 0, 2, 0, 3, 0, 4}},
b=9,{{0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 1, 2, 3, 4, 6}, {0, 0, 1, 2, 3, 5, 0, 1, 3}, {0, 0, 1, 2, 4, 0, 2, 4, 7}, {0, 1, 2, 0, 1, 3, 5, 0, 3}, {0, 1, 0, 1, 3, 0, 2, 5, 0}, {0, 1, 0, 2, 4, 1, 4, 0, 4}, {0, 1, 0, 2, 0, 3, 0, 4, 0}},
b=10,{{0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 1, 2, 3, 4, 5, 7}, {0, 0, 0, 1, 2, 3, 5, 7, 0, 2}, {0, 0, 1, 2, 4, 0, 1, 3, 5, 8}, {0, 1, 2, 0, 1, 3, 5, 0, 2, 5}, {0, 1, 2, 0, 2, 4, 0, 2, 5, 9}, {0, 1, 0, 1, 3, 0, 2, 5, 0, 4}, {0, 1, 0, 2, 0, 2, 5, 1, 5, 0}, {0, 1, 0, 2, 0, 3, 0, 4, 0, 5}},
b=11,{{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 7}, {0, 0, 0, 1, 2, 3, 4, 6, 8, 0, 2}, {0, 0, 1, 2, 3, 5, 0, 1, 3, 5, 8}, {0, 0, 1, 2, 4, 0, 2, 4, 7, 0, 3}, {0, 1, 2, 0, 1, 3, 5, 0, 2, 5, 9},
          {0, 1, 2, 0, 2, 4, 0, 3, 6, 0, 4}, {0, 1, 0, 1, 3, 0, 3, 6, 1, 5, 10}, {0, 1, 0, 2, 0, 2, 5, 1, 5, 0, 5}, {0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0}},
b=12,{{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8}, {0, 0, 0, 1, 2, 3, 4, 6, 8, 0, 1, 3}, {0, 0, 1, 2, 3, 5, 0, 1, 3, 5, 7, 10}, {0, 0, 1, 2, 4, 0, 1, 3, 5, 8, 0, 3}, {0, 1, 2, 0, 1, 3, 5, 0, 2, 5, 8, 0},
          {0, 1, 2, 0, 1, 3, 6, 1, 4, 7, 0, 4}, {0, 1,0, 1, 3, 0, 2, 5, 0, 3, 7, 0}, {0, 1, 0, 2, 4, 1, 4, 0, 3, 7, 1, 6}, {0, 1, 0, 2, 0, 3, 6, 2, 6, 1, 6, 0}, {0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6}},
b=13,{{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8}, {0, 0, 0, 0, 1, 2, 3, 4, 6, 8, 10, 0, 2}, {0, 0, 0, 1, 2, 3, 5, 7, 0, 2, 4, 6, 9}, {0, 0, 1, 2, 3, 5, 0, 2, 4, 6, 9, 0, 3}, {0, 0, 1, 2, 4, 0, 2, 4, 7, 0, 3, 6, 10},
          {0, 1, 2, 0, 1, 3, 5, 0, 2, 5, 8, 0, 4}, {0, 1, 2, 0, 2, 4, 0, 2, 5, 9, 2, 6, 11}, {0, 1, 0, 1, 3, 0, 2, 5, 0, 3, 7, 0, 5}, {0, 1, 0, 2, 4, 1, 4, 0, 3, 7, 1, 6, 12}, {0, 1, 0, 2, 0, 3, 6, 2, 6, 1, 6, 0, 6}, {0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0}},
b=14,{{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 9}, {0, 0, 0, 0, 1, 2, 3, 4, 5, 7, 9, 11, 0, 2}, {0, 0, 0, 1, 2, 3, 5, 7, 0, 1, 3, 5, 7, 10}, {0, 0, 1, 2, 3, 5, 0, 1, 3, 5, 7, 10, 0, 3},
          {0, 0, 1, 2, 4, 0, 2, 4, 6, 9, 1, 4, 7, 11}, {0, 1, 2, 0, 1, 3, 5, 0, 2, 5, 8, 0, 3, 7}, {0, 1, 2, 0, 1, 3, 6, 1, 4, 7, 0, 3, 7, 12}, {0, 1, 2, 0, 2, 4, 0, 3, 6, 0, 4, 8, 0, 5}, {0, 1, 0, 1, 3, 0, 3, 6, 1, 5, 9, 2, 7, 13},
          {0, 1, 0, 2, 4, 1, 4, 0, 4, 8, 2, 7, 0, 6}, {0, 1, 0, 2, 0, 3, 0, 3, 7, 2, 7, 1, 7, 0}, {0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7}},
b=15, {{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 9}, {0, 0, 0, 0, 1, 2, 3, 4, 5, 7, 9, 11, 0, 1, 3}, {0, 0, 0, 1, 2, 3, 4, 6, 8, 0, 1, 3, 5, 7, 10}, {0, 0, 1, 2, 3, 5, 0, 1, 3, 5, 7, 10, 0, 2, 5},
           {0, 0, 1, 2, 4, 0, 1, 3, 5, 8, 0, 2, 5, 8, 12}, {0, 0, 1, 2, 4, 0, 2, 4, 7, 0, 3, 6, 10, 0, 4}, {0, 1, 2, 0, 1, 3, 5, 0, 2, 5, 8, 0, 3, 7, 12}, {0, 1, 2, 0, 2, 4, 0, 2, 5, 9, 2, 6, 10, 1, 6}, {0, 1, 0, 1, 3, 0, 2, 5, 0, 3, 7, 0, 4, 9, 0},
           {0, 1, 0, 1, 3, 0, 3, 6, 1, 5, 10, 3, 8, 0, 6}, {0, 1, 0, 2, 0, 2, 5, 1, 5, 0, 4, 9, 2, 8, 0}, {0, 1, 0, 2, 0, 3, 0, 3, 7, 2, 7, 1, 7, 0, 7}, {0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0}},
b=16,{{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 10}, {0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 7, 9, 11, 13, 0, 2}, {0, 0, 0, 1, 2, 3, 4, 6, 8, 0, 1, 3, 5, 7, 9, 12}, {0, 0, 0, 1, 2, 3, 5, 7, 0, 2, 4, 6, 9, 12, 0, 3},
          {0, 0, 1, 2, 3, 5, 0, 2, 4, 6, 9, 0, 2, 5, 8, 12}, {0, 0, 1, 2, 4, 0, 2, 4, 6, 9, 1, 4, 7, 11, 0, 4}, {0, 1, 2, 0, 1, 3, 5, 0, 2, 5, 8, 0, 3, 7, 11, 0}, {0, 1, 2, 0, 1, 3, 5, 0, 3, 6, 10, 2, 6, 10, 0, 5}, {0, 1, 2, 0, 2, 4,  0, 3, 6, 0, 3, 7, 12, 3, 8, 14},
          {0, 1, 0, 1, 3, 0, 2, 5, 0, 3, 7, 0, 4, 9, 0, 6}, {0, 1, 0, 2, 4, 1, 4, 0, 3, 7, 1, 6, 11, 3, 9, 0}, {0, 1, 0, 2, 0, 2, 5, 1, 5, 0, 4, 9, 2, 8, 0, 7}, {0, 1, 0, 2, 0, 3, 0, 4, 8, 3, 8, 2, 8, 1, 8, 0}, {0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8}},
b=17,{{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 10}, {0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 0, 2}, {0, 0, 0, 0, 1, 2, 3, 4, 6, 8, 10, 0, 2, 4, 6, 8, 11}, {0, 0, 0, 1, 2, 3, 5, 7, 0, 1, 3, 5, 7, 10, 13, 0, 3},
          {0, 0, 1, 2, 3, 5, 0, 1, 3, 5, 7, 10, 0, 2, 5, 8, 12}, {0, 0, 1, 2, 4, 0, 1, 3, 5, 8, 0, 2, 5, 8, 12, 0, 4}, {0, 0, 1, 2, 4, 0, 2, 4, 7, 0, 3, 6, 10, 0, 4, 8, 13}, {0, 1, 2, 0, 1, 3, 5, 0, 2, 5, 8, 0, 3, 7, 11, 0, 5}, {0, 1, 2, 0, 1, 3, 6, 1, 4, 7, 0, 4, 8, 13, 3, 8, 14},
          {0, 1, 2, 0, 2, 4, 0, 3, 6, 0, 4, 8, 0, 5, 10, 0, 6}, {0, 1, 0, 1, 3, 0, 2, 5, 0, 4, 8, 1, 6, 11, 2, 8, 15}, {0, 1, 0, 2, 4, 1, 4, 0, 3, 7, 1, 6, 11, 3, 9, 0, 7}, {0, 1, 0, 2, 0, 2, 5, 1, 5, 0, 5, 10, 3, 9, 1, 8, 16},
          {0, 1, 0, 2, 0, 3, 0, 4, 8, 3, 8, 2, 8, 1, 8, 0, 8}, {0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0}},
b=18,{{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11}, {0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 0, 1, 3}, {0, 0, 0, 0, 1, 2, 3, 4, 6, 8,10, 0, 1, 3, 5, 7, 9, 12},
          {0, 0, 0, 1, 2, 3, 4, 6, 8, 0, 2, 4, 6, 8, 11, 14, 0, 3}, {0, 0, 1, 2, 3, 5, 0, 1, 3, 5, 7, 10, 0, 2, 5, 8, 11, 15}, {0, 0, 1, 2, 3, 5, 0, 2, 4, 6, 9, 0, 3, 6, 9, 13, 0,  4}, {0, 0, 1, 2, 4, 0, 2, 4, 7, 0, 2, 5, 8, 12, 1, 5, 9, 14},
          {0, 1, 2, 0, 1, 3, 5, 0, 2, 5, 8, 0, 3, 7, 11, 0, 4, 9}, {0, 1, 2, 0, 1, 3, 5, 0, 3, 6, 10, 2, 6, 10, 0, 4, 9, 15}, {0, 1, 2, 0, 2, 4, 0, 2, 5, 9, 2, 6, 10, 1, 6, 11, 0, 6}, {0, 1, 0, 1, 3, 0, 2, 5, 0, 3, 7, 0, 4, 9, 0, 5, 11, 0},
          {0, 1, 0, 1, 3, 0, 3, 6, 1, 5, 9, 2, 7, 13, 4, 10, 0, 7}, {0, 1, 0, 2, 4, 1, 4, 0, 4, 8, 2, 7, 0, 5, 11, 2, 9, 17}, {0, 1, 0, 2, 0, 3, 6, 2, 6, 1, 6, 0, 5, 11, 3, 10, 1, 9}, {0, 1, 0, 2, 0, 3, 0, 4, 0, 4, 9, 3, 9, 2, 9, 1, 9, 0}, {0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 9}},
b=19,{{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11}, {0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 0, 2}, {0, 0, 0, 0, 1, 2, 3, 4, 5, 7, 9, 11, 0, 1, 3, 5, 7, 9, 12},
          {0, 0, 0, 1, 2, 3, 4, 6, 8, 0, 1, 3, 5, 7, 9, 12, 15, 0, 3}, {0, 0, 0, 1, 2, 3, 5, 7, 0, 2, 4, 6, 9, 12, 0, 3, 6, 9, 13}, {0, 0, 1, 2, 3, 5, 0, 1, 3, 5, 8, 11, 1, 4, 7, 10, 14, 0, 4}, {0, 0, 1, 2, 4, 0, 1, 3, 5, 8, 0, 3, 6, 9, 13, 1, 5, 9, 14},
          {0, 0, 1, 2, 4, 0, 2, 4, 7, 0, 3, 6, 10, 0, 4, 8, 13, 0, 5}, {0, 1, 2, 0, 1, 3, 5, 0, 2, 5, 8, 0, 3, 7, 11, 0, 4, 9, 15}, {0, 1, 2, 0, 1, 3, 6, 1, 4, 7, 0, 3, 7, 12, 2, 7, 12, 0, 6}, {0, 1, 2, 0, 2, 4, 0, 3, 6, 0, 3, 7, 12, 3, 8, 14, 3, 9, 16},
          {0, 1, 0, 1, 3, 0, 2, 5, 0, 3, 7, 0, 4, 9, 0, 5, 11, 0, 7}, {0, 1, 0, 1, 3, 0, 3, 6, 1, 5, 10, 3, 8, 0, 6, 12, 2, 9, 17}, {0, 1, 0, 2, 4, 1, 4, 0, 4, 8, 2, 7, 0, 6, 12, 3, 10, 0, 8}, {0, 1, 0, 2, 0, 3, 6, 2, 6, 1, 6, 0, 5, 11, 3, 10, 1, 9, 18},
          {0, 1, 0, 2, 0, 3, 0, 4, 0, 4, 9, 3, 9, 2, 9, 1, 9, 0, 9}, {0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 9, 0}},

1. Table[Mod[Sum[Floor[k (a/b + 2 p)], {k, n}], n], {b, 2, 19}, {a, b - 1}, {n, b}]

复制代码

同上。

1. Table[Mod[Sum[Floor[k (a/b - 2 p)], {k, n}], n], {b, 2, 19}, {a, b - 1}, {n, b}]

复制代码

同上。

cgl_74

我的一个解法：
思路是分整数和非整数分类讨论。当为整数是，只有偶数能满足条件。为非整数时，拆分为整数部分和小数部分，可以证明这部分无解。
所以全部解就是所有偶数。