数字串的通项公式

王守恩 · 发表于 2025-6-26 10:17:48

n=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65,66, ......

\(\D\bigg\lfloor\sum_{k = 1}^{\infty}\frac{1}{(k^2+k)^{\frac{n+1}{2n}}}\bigg\rfloor=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, \cdots\cdots\)

王守恩 · 发表于 2025-6-26 10:45:19

n=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65,66, ......

\(\D\bigg\lfloor\sum_{k = 1}^{\infty}\frac{1}{k^{\frac{n+1}{n}}}\bigg\rfloor=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55,56,57, \cdots\cdots\)

王守恩 · 发表于 2025-6-26 17:04:21

n=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65,66, ......

\(\D\bigg\lfloor\sum_{k = 1}^{\infty}\frac{k^{\frac{n}{n+1}}}{k(k+1)}\bigg\rfloor=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55,56,\cdots\cdots\)

王守恩 · 发表于 2025-6-26 17:05:42

A=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, ...... n=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33 , ......

\(\D\bigg\lfloor\sum_{k=1}^{\infty}\frac{1}{(k^{A+1}+k^{A})^{\frac{n+1}{(A+1)n}}}\bigg\rfloor=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, \cdots\cdots\)

王守恩 · 发表于 7 天前

楼上的没问题！大胆往前走！！

\(\displaystyle\bigg\lfloor\sum_{k=1}^{\infty}\frac{1}{(k^{A+1}+k^{A})^{\frac{n+1}{n(A+1)}}}\bigg\rfloor=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48,\cdots\cdots\)

{1.000, 2.011, 3.014, 4.016, 5.016, 6.017, 7.017, 8.017, 9.018, 10.02, 11.02, 12.02, 13.02, 14.02, 15.02, 16.02, 17.02, 18.02, 19.02, 20.02, 21.02, 22.02, 23.02, 24.02, 25.02, 26.02, 27.02, 28.02, 29.02, 30.02, 31.02, 32.02},
{1.173, 2.184, 3.187, 4.188, 5.189, 6.189, 7.189, 8.189, 9.190, 10.19, 11.19, 12.19, 13.19, 14.19, 15.19, 16.19, 17.19, 18.19, 19.19, 20.19, 21.19, 22.19, 23.19, 24.19, 25.19, 26.19, 27.19, 28.19, 29.19, 30.19, 31.19, 32.19},
{1.274, 2.280, 3.281, 4.281, 5.282, 6.282, 7.282, 8.282, 9.282, 10.28, 11.28, 12.28, 13.28, 14.28, 15.28, 16.28, 17.28, 18.28, 19.28, 20.28, 21.28, 22.28, 23.28, 24.28, 25.28, 26.28, 27.28, 28.28, 29.28, 30.28, 31.28, 32.28},
{1.339, 2.341, 3.341, 4.340, 5.340, 6.339, 7.339, 8.339, 9.339, 10.34, 11.34, 12.34, 13.34, 14.34, 15.34, 16.34, 17.34, 18.34, 19.34, 20.34, 21.34, 22.34, 23.34, 24.34, 25.34, 26.34, 27.34, 28.34, 29.34, 30.34, 31.34, 32.34},
{1.385, 2.383, 3.381, 4.380, 5.380, 6.379, 7.379, 8.378, 9.378, 10.38, 11.38, 12.38, 13.38, 14.38, 15.38, 16.38, 17.38, 18.38, 19.38, 20.38, 21.38, 22.38, 23.38, 24.38, 25.38, 26.38, 27.38, 28.38, 29.38, 30.38, 31.38, 32.38},
{1.419, 2.414, 3.411, 4.409, 5.408, 6.408, 7.407, 8.407, 9.406, 10.41, 11.41, 12.41, 13.41, 14.41, 15.41, 16.41, 17.41, 18.41, 19.41, 20.40, 21.40, 22.40, 23.40, 24.40, 25.40, 26.40, 27.40, 28.40, 29.40, 30.40, 31.40, 32.40},
{1.445, 2.437, 3.434, 4.432, 5.430, 6.430, 7.429, 8.428, 9.428, 10.43, 11.43, 12.43, 13.43, 14.43, 15.43, 16.43, 17.43, 18.43, 19.43, 20.43, 21.43, 22.43, 23.43, 24.43, 25.43, 26.43, 27.43, 28.43, 29.43, 30.43, 31.43, 32.43},
{1.466, 2.456, 3.452, 4.449, 5.448, 6.447, 7.446, 8.445, 9.445, 10.44, 11.44, 12.44, 13.44, 14.44, 15.44, 16.44, 17.44, 18.44, 19.44, 20.44, 21.44, 22.44, 23.44, 24.44, 25.44, 26.44, 27.44, 28.44, 29.44, 30.44, 31.44, 32.44},
{1.483, 2.471, 3.466, 4.463, 5.462, 6.461, 7.460, 8.459, 9.459, 10.46, 11.46, 12.46, 13.46, 14.46, 15.46, 16.46, 17.46, 18.46, 19.46, 20.46, 21.46, 22.46, 23.46, 24.46, 25.46, 26.46, 27.46, 28.46, 29.46, 30.46, 31.46, 32.46},
{1.497, 2.483, 3.478, 4.475, 5.473, 6.472, 7.471, 8.470, 9.470, 10.47, 11.47, 12.47, 13.47, 14.47, 15.47, 16.47, 17.47, 18.47, 19.47, 20.47, 21.47, 22.47, 23.47, 24.47, 25.47, 26.47, 27.47, 28.47, 29.47, 30.47, 31.47, 32.47},
{1.508, 2.494, 3.488, 4.485, 5.483, 6.481, 7.480, 8.480, 9.479, 10.48, 11.48, 12.48, 13.48, 14.48, 15.48, 16.48, 17.48, 18.48, 19.48, 20.48, 21.48, 22.48, 23.48, 24.48, 25.48, 26.48, 27.48, 28.48, 29.48, 30.48, 31.48, 32.48},
{1.518, 2.502, 3.496, 4.493, 5.491, 6.490, 7.489, 8.488, 9.487, 10.49, 11.49, 12.49, 13.49, 14.49, 15.49, 16.49, 17.48, 18.48, 19.48, 20.48, 21.48, 22.48, 23.48, 24.48, 25.48, 26.48, 27.48, 28.48, 29.48, 30.48, 31.48, 32.48},
{1.527, 2.510, 3.504, 4.500, 5.498, 6.496, 7.495, 8.495, 9.494, 10.49, 11.49, 12.49, 13.49, 14.49, 15.49, 16.49, 17.49, 18.49, 19.49, 20.49, 21.49, 22.49, 23.49, 24.49, 25.49, 26.49, 27.49, 28.49, 29.49, 30.49, 31.49, 32.49},
{1.534, 2.517, 3.510, 4.506, 5.504, 6.503, 7.501, 8.501, 9.500, 10.50, 11.50, 12.50, 13.50, 14.50, 15.50, 16.50, 17.50, 18.50, 19.50, 20.50, 21.50, 22.50, 23.50, 24.50, 25.50, 26.50, 27.50, 28.50, 29.50, 30.50, 31.50, 32.50},
{1.541, 2.522, 3.515, 4.512, 5.509, 6.508, 7.507, 8.506, 9.505, 10.50, 11.50, 12.50, 13.50, 14.50, 15.50, 16.50, 17.50, 18.50, 19.50, 20.50, 21.50, 22.50, 23.50, 24.50, 25.50, 26.50, 27.50, 28.50, 29.50, 30.50, 31.50, 32.50},
{1.547, 2.528, 3.520, 4.517, 5.514, 6.513, 7.511, 8.511, 9.510, 10.51, 11.51, 12.51, 13.51, 14.51, 15.51, 16.51, 17.51, 18.51, 19.51, 20.51, 21.51, 22.51, 23.51, 24.51, 25.51, 26.51, 27.51, 28.51, 29.51, 30.51, 31.51, 32.51},
{1.552, 2.532, 3.525, 4.521, 5.518, 6.517, 7.516, 8.515, 9.514, 10.51, 11.51, 12.51, 13.51, 14.51, 15.51, 16.51, 17.51, 18.51, 19.51, 20.51, 21.51, 22.51, 23.51, 24.51, 25.51, 26.51, 27.51, 28.51, 29.51, 30.51, 31.51, 32.51},
{1.557, 2.536, 3.529, 4.525, 5.522, 6.520, 7.519, 8.518, 9.518, 10.52, 11.52, 12.52, 13.52, 14.52, 15.52, 16.52, 17.51, 18.51, 19.51, 20.51, 21.51, 22.51, 23.51, 24.51, 25.51, 26.51, 27.51, 28.51, 29.51, 30.51, 31.51, 32.51},
{1.561, 2.540, 3.532, 4.528, 5.526, 6.524, 7.523, 8.522, 9.521, 10.52, 11.52, 12.52, 13.52, 14.52, 15.52, 16.52, 17.52, 18.52, 19.52, 20.52, 21.52, 22.52, 23.52, 24.52, 25.52, 26.52, 27.52, 28.52, 29.52, 30.52, 31.52, 32.52},
{1.565, 2.543, 3.535, 4.531, 5.529, 6.527, 7.526, 8.525, 9.524, 10.52, 11.52, 12.52, 13.52, 14.52, 15.52, 16.52, 17.52, 18.52, 19.52, 20.52, 21.52, 22.52, 23.52, 24.52, 25.52, 26.52, 27.52, 28.52, 29.52, 30.52, 31.52, 32.52},
{1.569, 2.547, 3.538, 4.534, 5.531, 6.530, 7.528, 8.528, 9.527, 10.53, 11.53, 12.53, 13.52, 14.52, 15.52, 16.52, 17.52, 18.52, 19.52, 20.52, 21.52, 22.52, 23.52, 24.52, 25.52, 26.52, 27.52, 28.52, 29.52, 30.52, 31.52, 32.52},
{1.572, 2.549, 3.541, 4.537, 5.534, 6.532, 7.531, 8.530, 9.529, 10.53, 11.53, 12.53, 13.53, 14.53, 15.53, 16.53, 17.53, 18.53, 19.53, 20.53, 21.53, 22.53, 23.53, 24.53, 25.53, 26.53, 27.53, 28.53, 29.53, 30.52, 31.52, 32.52},
{1.575, 2.552, 3.543, 4.539, 5.536, 6.535, 7.533, 8.532, 9.532, 10.53, 11.53, 12.53, 13.53, 14.53, 15.53, 16.53, 17.53, 18.53, 19.53, 20.53, 21.53, 22.53, 23.53, 24.53, 25.53, 26.53, 27.53, 28.53, 29.53, 30.53, 31.53, 32.53},
{1.578, 2.554, 3.546, 4.541, 5.539, 6.537, 7.535, 8.534, 9.534, 10.53, 11.53, 12.53, 13.53, 14.53, 15.53, 16.53, 17.53, 18.53, 19.53, 20.53, 21.53, 22.53, 23.53, 24.53, 25.53, 26.53, 27.53, 28.53, 29.53, 30.53, 31.53, 32.53},
{1.580, 2.556, 3.548, 4.543, 5.541, 6.539, 7.537, 8.536, 9.536, 10.54, 11.53, 12.53, 13.53, 14.53, 15.53, 16.53, 17.53, 18.53, 19.53, 20.53, 21.53, 22.53, 23.53, 24.53, 25.53, 26.53, 27.53, 28.53, 29.53, 30.53, 31.53, 32.53},
{1.582, 2.559, 3.550, 4.545, 5.542, 6.541, 7.539, 8.538, 9.537, 10.54, 11.54, 12.54, 13.54, 14.54, 15.53, 16.53, 17.53, 18.53, 19.53, 20.53, 21.53, 22.53, 23.53, 24.53, 25.53, 26.53, 27.53, 28.53, 29.53, 30.53, 31.53, 32.53},
{1.585, 2.560, 3.552, 4.547, 5.544, 6.542, 7.541, 8.540, 9.539, 10.54, 11.54, 12.54, 13.54, 14.54, 15.54, 16.54, 17.54, 18.54, 19.54, 20.54, 21.54, 22.54, 23.54, 24.54, 25.54, 26.53, 27.53, 28.53, 29.53, 30.53, 31.53, 32.53},
{1.587, 2.562, 3.553, 4.549, 5.546, 6.544, 7.543, 8.542, 9.541, 10.54, 11.54, 12.54, 13.54, 14.54, 15.54, 16.54, 17.54, 18.54, 19.54, 20.54, 21.54, 22.54, 23.54, 24.54, 25.54, 26.54, 27.54, 28.54, 29.54, 30.54, 31.54, 32.54},
{1.589, 2.564, 3.555, 4.550, 5.547, 6.545, 7.544, 8.543, 9.542, 10.54, 11.54, 12.54, 13.54, 14.54, 15.54, 16.54, 17.54, 18.54, 19.54, 20.54, 21.54, 22.54, 23.54, 24.54, 25.54, 26.54, 27.54, 28.54, 29.54, 30.54, 31.54, 32.54},
{1.590, 2.565, 3.556, 4.552, 5.549, 6.547, 7.545, 8.544, 9.544, 10.54, 11.54, 12.54, 13.54, 14.54, 15.54, 16.54, 17.54, 18.54, 19.54, 20.54, 21.54, 22.54, 23.54, 24.54, 25.54, 26.54, 27.54, 28.54, 29.54, 30.54, 31.54, 32.54},
{1.592, 2.567, 3.558, 4.553, 5.550, 6.548, 7.547, 8.546, 9.545, 10.54, 11.54, 12.54, 13.54, 14.54, 15.54, 16.54, 17.54, 18.54, 19.54, 20.54, 21.54, 22.54, 23.54, 24.54, 25.54, 26.54, 27.54, 28.54, 29.54, 30.54, 31.54, 32.54},
{1.594, 2.568, 3.559, 4.554, 5.551, 6.549, 7.548, 8.547, 9.546, 10.55, 11.54, 12.54, 13.54, 14.54, 15.54, 16.54, 17.54, 18.54, 19.54, 20.54, 21.54, 22.54, 23.54, 24.54, 25.54, 26.54, 27.54, 28.54, 29.54, 30.54, 31.54, 32.54},
{1.595, 2.569, 3.560, 4.555, 5.553, 6.551, 7.549, 8.548, 9.547, 10.55, 11.55, 12.55, 13.55, 14.54, 15.54, 16.54, 17.54, 18.54, 19.54, 20.54, 21.54, 22.54, 23.54, 24.54, 25.54, 26.54, 27.54, 28.54, 29.54, 30.54, 31.54, 32.54},
{1.596, 2.571, 3.561, 4.557, 5.554, 6.552, 7.550, 8.549, 9.548, 10.55, 11.55, 12.55, 13.55, 14.55, 15.55, 16.55, 17.55, 18.54, 19.54, 20.54, 21.54, 22.54, 23.54, 24.54, 25.54, 26.54, 27.54, 28.54, 29.54, 30.54, 31.54, 32.54},
{1.598, 2.572, 3.562, 4.558, 5.555, 6.553, 7.551, 8.550, 9.549, 10.55, 11.55, 12.55, 13.55, 14.55, 15.55, 16.55, 17.55, 18.55, 19.55, 20.55, 21.55, 22.55, 23.55, 24.55, 25.54, 26.54, 27.54, 28.54, 29.54, 30.54, 31.54, 32.54},
{1.599, 2.573, 3.563, 4.559, 5.556, 6.554, 7.552, 8.551, 9.550, 10.55, 11.55, 12.55, 13.55, 14.55, 15.55, 16.55, 17.55, 18.55, 19.55, 20.55, 21.55, 22.55, 23.55, 24.55, 25.55, 26.55, 27.55, 28.55, 29.55, 30.55, 31.55, 32.55},
{1.600, 2.574, 3.564, 4.560, 5.557, 6.555, 7.553, 8.552, 9.551, 10.55, 11.55, 12.55, 13.55, 14.55, 15.55, 16.55, 17.55, 18.55, 19.55, 20.55, 21.55, 22.55, 23.55, 24.55, 25.55, 26.55, 27.55, 28.55, 29.55, 30.55, 31.55, 32.55},
{1.601, 2.575, 3.565, 4.560, 5.557, 6.555, 7.554, 8.553, 9.552, 10.55, 11.55, 12.55, 13.55, 14.55, 15.55, 16.55, 17.55, 18.55, 19.55, 20.55, 21.55, 22.55, 23.55, 24.55, 25.55, 26.55, 27.55, 28.55, 29.55, 30.55, 31.55, 32.55},
{1.602, 2.576, 3.566, 4.561, 5.558, 6.556, 7.555, 8.554, 9.553, 10.55, 11.55, 12.55, 13.55, 14.55, 15.55, 16.55, 17.55, 18.55, 19.55, 20.55, 21.55, 22.55, 23.55, 24.55, 25.55, 26.55, 27.55, 28.55, 29.55, 30.55, 31.55, 32.55},
{1.603, 2.577, 3.567, 4.562, 5.559, 6.557, 7.556, 8.555, 9.554, 10.55, 11.55, 12.55, 13.55, 14.55, 15.55, 16.55, 17.55, 18.55, 19.55, 20.55, 21.55, 22.55, 23.55, 24.55, 25.55, 26.55, 27.55, 28.55, 29.55, 30.55, 31.55, 32.55}}

Table[N[Sum[{1/((k^(A + 1) + k^A)^((n + 1)/(n (A + 1))))}, {k, Infinity}], 4], {A, 40}, {n, 32}]

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王守恩 · 发表于前天 09:01

A344183——蛮好玩的一串数。
1 1
2 12
3 123
4 1234
5 12345
6 123456
7 1234567
8 12345678
9 123456789
10 1023456789
11 11023456789
12 110123456789
13 1101213456789
14 11012131456789
15 110121314156789
16 1101213141516789
17 11012131415161789
18 110121314151617189
19 1101213141516171819
20 11012013141516171819
21 110120131415161718219
22 1101201314151617182219
23 110120131415161718221923
24 11012013141516171822192324
25 1101201314151617182219232425
26 110120131415161718221923242526
27 11012013141516171822192324252627
28 1101201314151617182219232425262728
29 110120131415161718221923242526272829
30 1101201301415161718221923242526272829
31 11012013014151617182219231242526272829
32 11032013014151617182219231242526272829
33 110320130141516171822192331242526272829
34 11032013014151617182219233124252627282934
35 1103201301415161718221923312425262728293435
36 110320130141516171822192331242526272829343536

王守恩 · 发表于前天 18:56

aimisiyou 发表于 2025-6-21 09:25
从1~100中至少选出多少个不同的数，必然存在四个不同的数，满足a+b=c+d?

接715#,716#——记录一下手工解题的过程。

题目。{1, 2, ..., n}没有等和对的最大子集a(k), n>3。

a(3)=3, {1,2,3}——每次只要关注末尾2个数之间数是否有解就可以了。譬如:
a(4)=5, {1,2,3,5}——3,5之间有4——1,2,3,4有解。
a(5)=8, {1,2,3,5,8}——5,8之间有6,7——1,2,3,5,6-1,2,3,5,7有解。
a(6)=13, {1,2,3,5,8,13}——8,13之间有9,10,11,12——1,2,3,5,8,9-1,2,3,5,8,12有解。
a(7)=21, {1,2,3,5,8,13,21}——13,21之间有14-20——1,2,3,5,8,13,14-1,2,3,5,8,13,20有解。
a(8)=30, {1,2,3,5,8,13,21,30}——21,30之间有22-29——1,2,3,5,8,13,21,22-1,2,3,5,8,13,21,29有解。
a(9)=39, {1,2,3,5,8,13,21,30,39}——30,39之间有31-38——1,2,3,5,8,13,21,30,31-1,2,3,5,8,13,21,38有解。
a(10)=53, {1,2,3,5,8,13,21,30,39,53}
a(11)=74, {1,2,3,5,8,13,21,30,39,53,74},
a(12)=95, {1,2,3,5,8,13,21,30,39,53,74,95},

得到——A011185——1, 2, 3, 5, 8, 13, 21, 30, 39, 53, 74, 95, 128, 152, 182, 212, 258, 316, 374, 413, 476, 531, 546, 608, 717, 798, 862, 965, 1060, 1161, 1307, 1386, 1435, 1556, 1722, 1834, 1934, 2058, ......

这个通项公式也可以。

t = {1}; k = 1; sms = {}; Do[k++; While[Intersection[sms, t + k] != {}, k++]; sms = Join[sms, t + k, {k}]; AppendTo[t, k], {50}]; t

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又：{1, 2, ..., n}没有等和对的最大子集a(k)可以有很多数字串——只要把这个"A"换一下就行。

t = {1}; k = 1; sms = {}; Do[k++; While[Intersection[sms, t + k] != {}, k++]; sms = Join[sms, t + k, {A*k}]; AppendTo[t, k], {50}]; t

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[原创] 数字串的通项公式

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