数字串的通项公式

northwolves · 发表于 2025-1-6 18:03:36

本帖最后由 northwolves 于 2025-1-7 07:59 编辑

王守恩发表于 2025-1-6 15:51
A008904——a(n) is the final nonzero digit of n!.

1, 1, 2, 6, 4, 2, 2, 4, 2, 8, 8, 8, 6, 8, 2, 8, 8 ...

Table[Last[IntegerDigits[n!] /. 0 -> Nothing], {n, 0, 25}]

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Table[Select[IntegerDigits[n!], # > 0 &][[-1]], {n, 0, 25}]

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Table[Drop[IntegerDigits[n!], -Sum[Floor[n/5^k], {k, Log[5, n]}]][[-1]], {n, 0, 25}]

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王守恩 · 发表于 2025-1-7 10:25:39

本帖最后由王守恩于 2025-1-7 16:00 编辑

northwolves 发表于 2025-1-6 18:03

1,(1)1,(2)2,(3)6,(4)4,(3)2,(1)2,(2)4,(3)2,(4)8,(1)
8,(1)8,(2)6,(3)8,(4)2,(4)8,(1)8,(2)6,(3)8,(4)2,(2)
4,(1)4,(2)8,(3)4,(4)6,(4)4,(1)4,(2)8,(3)4,(4)6,(3)
8,(1)8,(2)6,(3)8,(4)2,(1)2,(1)2,(2)4,(3)2,(4)8,(4)
2,(1)2,(2)4,(3)2,(4)8,(2)6,(1)6,(2)2,(3)6,(4)4,(3)
2,(1)2,(2)4,(3)2,(4)8,(3)4,(1)4,(2)8,(3)4,(4)6,(1)
6,(1)6,(2)2,(3)6,(4)4,(4)6,(1)6,(2)2,(3)6,(4)4,(2)
8,(1)8,(2)6,(3)8,(4)2,(2)4,(1)4,(2)8,(3)4,(4)6,(3)
8,(1)8,(2)6,(3)8,(4)2,(1)2,(1)2,(2)4,(3)2,(4)8,(4)
2,(1)2,(2)4,(3)2,(4)8,(2)6,(1)6,(2)2,(3)6,(4)4,(1)
4,(1)4,(2)8,(3)4,(4)6,(3)8,(1)8,(2)6,(3)8,(4)2,(1)
2,(1)2,(2)4,(3)2,(4)8,(4)2,(1)2,(2)4,(3)2,(4)8,(2)
6,(1)6,(2)2,(3)6,(4)4,(2)8,(1)8,(2)6,(3)8,(4)2,(3)
6,(1)6,(2)2,(3)6,(4)4,(1)4,(1)4,(2)8,(3)4,(4)6,(4)
4,(1)4,(2)8,(3)4,(4)6,(2)2,(1)2,(2)4,(3)2,(4)8,(4)
2,(1)2,(2)4,(3)2,(4)8,(3)4,(1)4,(2)8,(3)4,(4)6,(1)
6,(1)6,(2)2,(3)6,(4)4,(4)6,(1)6,(2)2,(3)6,(4)4,(2)
8,(1)8,(2)6,(3)8,(4)2,(3)6,(1)6,(2)2,(3)6,(4)4,(3)
2,(1)2,(2)4,(3)2,(4)8,(1)8,(1)8,(2)6,(3)8,(4)2,(4)
8,(1)8,(2)6,(3)8,(4)2,(2)4,(1)4,(2)8,(3)4,(4)6,(2)

1, 1, 2, 6, 4,(3)2, 2, 4, 2, 8,(1)
8, 8, 6, 8, 2,(4)8, 8, 6, 8, 2,(2)
4, 4, 8, 4, 6,(4)4, 4, 8, 4, 6,(3)
8, 8, 6, 8, 2,(1)2, 2, 4, 2, 8,(4)
2, 2, 4, 2, 8,(2)6, 6, 2, 6, 4,(3)
2, 2, 4, 2, 8,(3)4, 4, 8, 4, 6,(1)
6, 6, 2, 6, 4,(4)6, 6, 2, 6, 4,(2)
8, 8, 6, 8, 2,(2)4, 4, 8, 4, 6,(3)
8, 8, 6, 8, 2,(1)2, 2, 4, 2, 8,(4)
2, 2, 4, 2, 8,(2)6, 6, 2, 6, 4,(1)
4, 4, 8, 4, 6,(3)8, 8, 6, 8, 2,(1)
2, 2, 4, 2, 8,(4)2, 2, 4, 2, 8,(2)
6, 6, 2, 6, 4,(2)8, 8, 6, 8, 2,(3)
6, 6, 2, 6, 4,(1)4, 4, 8, 4, 6,(4)
4, 4, 8, 4, 6,(2)2, 2, 4, 2, 8,(4)
2, 2, 4, 2, 8,(3)4, 4, 8, 4, 6,(1)
6, 6, 2, 6, 4,(4)6, 6, 2, 6, 4,(2)
8, 8, 6, 8, 2,(3)6, 6, 2, 6, 4,(3)
2, 2, 4, 2, 8,(1)8, 8, 6, 8, 2,(4)
8, 8, 6, 8, 2,(2)4, 4, 8, 4, 6,(2)
2, 2, 4, 2, 8,(3)4, 4, 8, 4, 6,(1)
6, 6, 2, 6, 4,(4)6, 6, 2, 6, 4,(2)
8, 8, 6, 8, 2,(1)2, 2, 4, 2, 8,(3)
4, 4, 8, 4, 6,(1)6, 6, 2, 6, 4,(4)
6, 6, 2, 6, 4,(2)8, 8, 6, 8, 2,(4)
8, 8, 6, 8, 2,(3)6, 6, 2, 6, 4,(1)
4, 4, 8, 4, 6,(4)4, 4, 8, 4, 6,(2)
2, 2, 4, 2, 8,(4)2, 2, 4, 2, 8,(3)
4, 4, 8, 4, 6,(1)6, 6, 2, 6, 4,(4)
6, 6, 2, 6, 4,(2)8, 8, 6, 8, 2,(3)
6, 6. 2, 6, 4,(3)2, 2, 4, 2, 8,(1)
8, 8, 6, 8, 2,(4)8, 8, 6, 8, 2,(2)
4, 4, 8, 4, 6,(2)2, 2, 4, 2, 8,(3)
4, 4, 8, 4, 6,(1)6, 6, 2, 6, 4,(4)
6, 6, 2, 6, 4,(2)8, 8, 6, 8, 2,(1)
2, 2, 4, 2, 8,(3)4, 4, 8, 4, 6,(1)
6, 6, 2, 6, 4,(4)6, 6, 2, 6, 4,(2)
8, 8, 6, 8, 2,(1)2, 2, 4, 2, 8,(3)
4, 4, 8, 4, 6,(1)6, 6, 2, 6, 4,(4)
6, 6, 2, 6, 4,(2)8, 8, 6, 8, 2,(4)
8, 8, 6, 8, 2,(3)6, 6, 2, 6, 4,(1)
4, 4, 8, 4, 6,(4)4, 4, 8, 4, 6,(2)
2, 2, 4, 2, 8,(3)4, 4, 8, 4, 6,(3)
8, 8, 6, 8, 2,(1)2, 2, 4, 2, 8,(4)
2, 2, 4, 2, 8,(2)6, 6, 2, 6, 4,(2)
8, 8, 6, 8, 2,(3)6, 6, 2, 6, 4,(1)
4, 4, 8, 4, 6,(4)4, 4, 8, 4, 6,(2)
2, 2, 4, 2, 8,(1)8, 8, 6, 8, 2,(3)
6, 6, 2, 6, 4,(1)4, 4, 8, 4, 6,(4)
4, 4, 8, 4, 6,(2)2, 2, 4, 2, 8,(3)
4, 4, 8, 4, 6,(3)8, 8, 6, 8, 2,(1)
2, 2, 4, 2, 8,(4)2, 2, 4, 2, 8,(2)
6, 6, 2, 6, 4,(4)6, 6, 2, 6, 4,(3)
2, 2, 4, 2, 8,(1)8, 8, 6, 8, 2,(4)
8, 8, 6, 8, 2,(2)4, 4, 8, 4, 6,(3)
8, 8, 6, 8, 2,(3)6, 6, 2, 6, 4,(1)
4, 4, 8, 4, 6,(4)4, 4, 8, 4, 6,(2)
2, 2, 4, 2, 8,(2)6, 6, 2, 6, 4,(3)
2, 2, 4, 2, 8,(1)8, 8, 6, 8, 2,(4)
8, 8, 6, 8, 2,(2)4, 4, 8, 4, 6,(1)
6, 6, 2, 6, 4,(3)2, 2, 4, 2, 8,(1)
8, 8, 6, 8, 2,(4)8, 8, 6, 8, 2,(2)
4, 4, 8, 4, 6,(1)6, 6, 2, 6, 4,(3)
2, 2, 4, 2, 8,(1)8, 8, 6, 8, 2,(4)
8, 8, 6, 8, 2,(2)4, 4, 8, 4, 6,(4)
4, 4, 8, 4, 6,(3)8, 8, 6, 8, 2,(1)
2, 2, 4, 2, 8,(4)2, 2, 4, 2, 8,(2)
6, 6, 2, 6, 4,(3)2, 2, 4, 2, 8,(3)
4, 4, 8, 4, 6,(1)6, 6, 2, 6, 4,(4)
6, 6, 2, 6, 4,(2)8, 8, 6, 8, 2,(2)
4, 4, 8, 4, 6,(3)8, 8, 6, 8, 2,(1)
2, 2, 4, 2, 8,(4)2, 2, 4, 2, 8,(2)
6, 6, 2, 6, 4,(1)4, 4, 8, 4, 6,(3)
8, 8, 6, 8, 2,(1)2, 2, 4, 2, 8,(4)
2, 2, 4, 2, 8,(2)6, 6, 2, 6, 4,(2)
8, 8, 6, 8, 2,(3)6, 6, 2, 6, 4,(1)
4, 4, 8, 4, 6,(4)4, 4, 8, 4, 6,(2)
2, 2, 4, 2, 8,(4)2, 2, 4, 2, 8,(3)
4, 4, 8, 4, 6,(1)6, 6, 2, 6, 4,(4)
6, 6, 2, 6, 4,(2)8, 8, 6, 8, 2,(3)
6, 6, 2, 6, 4,(3)2, 2, 4, 2, 8,(1)
8, 8, 6, 8, 2,(4)8, 8, 6, 8, 2,(2)
4, 4, 8, 4, 6,(2)2, 2, 4, 2, 8,(3)
4, 4, 8, 4, 6,(1)6, 6, 2, 6, 4,(4)
6, 6, 2, 6, 4,(2)8, 8, 6, 8, 2,(1)
2, 2, 4, 2, 8,(3)4, 4, 8, 4, 6,(1)
6, 6, 2, 6, 4,(4)6, 6, 2, 6, 4,(2)
8, 8, 6, 8, 2,(4)8, 8, 6, 8, 2,(3)
6, 6, 2, 6, 4,(1)4, 4, 8, 4, 6,(4)
4, 4, 8, 4, 6,(2)2, 2, 4, 2, 8,(4)
2, 2, 4, 2, 8,(3)4, 4, 8, 4, 6,(1)
6, 6, 2, 6, 4,(4)6, 6, 2, 6, 4,(2)
8, 8, 6, 8, 2,(3)6, 6, 2, 6, 4,(3)
2, 2, 4, 2, 8,(1)8, 8, 6, 8, 2,(4)
8, 8, 6, 8, 2,(2)4, 4, 8, 4, 6,(2)
2, 2, 4, 2, 8,(3)4, 4, 8, 4, 6,(1)
6, 6, 2, 6, 4,(4)6, 6, 2, 6, 4,(2)
8, 8, 6, 8, 2,(1)2, 2, 4, 2, 8,(3)
4, 4, 8, 4, 6,(1)6, 6, 2, 6, 4,(4)
6, 6, 2, 6, 4,(2)8, 8, 6, 8, 2,()

还真是个无理数。

王守恩 · 发表于 2025-1-7 12:19:55

northwolves 发表于 2025-1-6 18:03

憋了好多天了。不敢问。R=2025。还有第2个解吗？

Solve[{(Sqrt[R (R - 2 c)] - c)/R == (2 b (Sqrt[R (R - 2 b)] - 2 b + R))/( b^2 + (Sqrt[R (R - 2 b)] - 2 b + R)^2), 2500 > R > c > b > 0}, {R, b, c}, Integers]

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知乎——2025 跨年优雅几何题, 来一个学习意志! 大家有没有更优雅的方法?

一道几乎没有数字的简洁几何题, 答案是 2025, 祝大家新年快乐!

王守恩 · 发表于 2025-1-14 16:48:35

A211264——整数对数{x, y}, 使得 0 < x < y ≤ n 和 x*y ≤ n。

0, 1, 2, 3, 4, 6, 7, 9, 10, 12, 13, 16, 17, 19, 21, 23, 24, 27, 28, 31, 33, 35, 36, 40, 41, 43, 45, 48, 49, 53, 54, 57, 59, 61, 63, 67, 68, 70, 72, 76, 77, 81, 82, 85, 88, 90, 91, 96, 97, 100, 102, 105, 106, 110, 112, 116, 118, 120, 121,

a(1)=0
a(2)=1={1,2},
a(3)=2={1,2},{1,3},
a(4)=3={1,2},{1,3},{1,4},
a(5)=4={1,2},{1,3},{1,4},{1,5},
a(6)=6={1,2},{1,3},{1,4},{1,5},{1,6},{2,3},
a(7)=7={1,2},{1,3},{1,4},{1,5},{1,6},{2,3},{1,7},
a(8)=9={1,2},{1,3},{1,4},{1,5},{1,6},{2,3},{1,7},{1,8},{2,4},
a(9)=10={1,2},{1,3},{1,4},{1,5},{1,6},{2,3},{1,7},{1,8},{2,4},{1,9},
a(10)=12={1,2},{1,3},{1,4},{1,5},{1,6},{2,3},{1,7},{1,8},{2,4},{1,9},{1,10},{2,5},
a(11)=13={1,2},{1,3},{1,4},{1,5},{1,6},{2,3},{1,7},{1,8},{2,4},{1,9},{1,10},{2,5},{1,11},
a(12)=16={1,2},{1,3},{1,4},{1,5},{1,6},{2,3},{1,7},{1,8},{2,4},{1,9},{1,10},{2,5},{1,11},{1,12},{2,6},{3,4},
a(13)=17={1,2},{1,3},{1,4},{1,5},{1,6},{2,3},{1,7},{1,8},{2,4},{1,9},{1,10},{2,5},{1,11},{1,12},{2,6},{3,4},{1,13},
a(14)=19={1,2},{1,3},{1,4},{1,5},{1,6},{2,3},{1,7},{1,8},{2,4},{1,9},{1,10},{2,5},{1,11},{1,12},{2,6},{3,4},{1,13},{1,14},{2,7},
a(15)=21={1,2},{1,3},{1,4},{1,5},{1,6},{2,3},{1,7},{1,8},{2,4},{1,9},{1,10},{2,5},{1,11},{1,12},{2,6},{3,4},{1,13},{1,14},{2,7},{1,15},{3,5},
a(16)=23={1,2},{1,3},{1,4},{1,5},{1,6},{2,3},{1,7},{1,8},{2,4},{1,9},{1,10},{2,5},{1,11},{1,12},{2,6},{3,4},{1,13},{1,14},{2,7},{1,15},{3,5},{1,16},{2,8},
a(17)=24={1,2},{1,3},{1,4},{1,5},{1,6},{2,3},{1,7},{1,8},{2,4},{1,9},{1,10},{2,5},{1,11},{1,12},{2,6},{3,4},{1,13},{1,14},{2,7},{1,15},{3,5},{1,16},{2,8},{1,17},
a(18)=27={1,2},{1,3},{1,4},{1,5},{1,6},{2,3},{1,7},{1,8},{2,4},{1,9},{1,10},{2,5},{1,11},{1,12},{2,6},{3,4},{1,13},{1,14},{2,7},{1,15},{3,5},{1,16},{2,8},{1,17},{1,18},{2,9},{3,6},
a(19)=28={1,2},{1,3},{1,4},{1,5},{1,6},{2,3},{1,7},{1,8},{2,4},{1,9},{1,10},{2,5},{1,11},{1,12},{2,6},{3,4},{1,13},{1,14},{2,7},{1,15},{3,5},{1,16},{2,8},{1,17},{1,18},{2,9},{3,6},{1,19},
a(20)=31,
Table[Sum[Floor[(n - k^2)/k], {k, Sqrt[n]}], {n, 75}]——这样不是更好？！！
Table[Sum[Floor[DivisorSigma[0, k]/2], {k, n}], {n, 75}]

northwolves · 发表于 2025-1-15 09:30:22

王守恩发表于 2025-1-14 16:48
A211264——整数对数{x, y}, 使得 0 < x < y ≤ n 和 x*y ≤ n。

0, 1, 2, 3, 4, 6, 7, 9, 10, 12, 13, 1 ...

Table[Sum[Floor[n/k] - k], {k, Sqrt[n]}], {n, 75}]

似乎这样确实更好些

王守恩 · 发表于 2025-1-15 11:20:42

northwolves 发表于 2025-1-15 09:30
Table[Sum[Floor[n/k] - k], {k, Sqrt[n]}], {n, 75}]

似乎这样确实更好些

A020897——只计算到——120。

Table[FindInstance[{x^3 + y^3 == z^3 n, 0 < y, 0 < z < 2000}, {x, y, z}, Integers, 1], {n, 120}]

我这些按钮不会用，你那里肯定不止120的。

补充内容 (2025-1-17 07:10):
过！我连51也出不来！

王守恩 · 发表于 2025-1-17 16:10:46

northwolves 发表于 2025-1-15 09:30
Table[Sum[Floor[n/k] - k], {k, Sqrt[n]}], {n, 75}]

似乎这样确实更好些

这样确实更好些——难怪OEIS没有了——奇数对数{x, y}, 使得 0 < x < y ≤ 2 n + 1 和 x*y ≤2 n + 1——这通项公式还不好找？！

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 22, 23, 25, 26, 27, 30, 31, 32, 34, 35, 37, 39, 40, 41, 44, 46, 47, 49, 50, 51, 54, 56, 57, 59, 60, 62, 64, 65, 67, 69, 71, 72, 74, 75

a(0)=0
a(1)=1={1,3},
a(2)=2={1,3},{1,5},
a(3)=3={1,3},{1,5},{1,7},
a(4)=4={1,3},{1,5},{1,7},{1,9},
a(5)=5={1,3},{1,5},{1,7},{1,9},{1,11},
a(6)=6={1,3},{1,5},{1,7},{1,9},{1,11},{1,13},
a(7)=8={1,3},{1,5},{1,7},{1,9},{1,11},{1,13},{1,15},{3,5},
a(8)=9={1,3},{1,5},{1,7},{1,6},{1,11},{1,13},{1,15},{3,5},{1,17},
a(9)=10={1,3},{1,5},{1,7},{1,9},{1,11},{1,13},{1,15},{3,5},{1,17},{1,19},
a(10)=12={1,3},{1,5},{1,7},{1,9},{1,11},{1,13},{1,15},{3,5},{1,17},{1,19},{1,21},{3,7},
a(11)=13={1,3},{1,5},{1,7},{1,9},{1,11},{1,13},{1,15},{3,5},{1,17},{1,19},{1,21},{3,7},{1,23},
a(12)=14={1,3},{1,5},{1,7},{1,9},{1,11},{1,13},{1,15},{3,5},{1,17},{1,19},{1,21},{3,7},{1,23},{1,25},
a(13)=16={1,3},{1,5},{1,7},{1,9},{1,11},{1,13},{1,15},{3,5},{1,17},{1,19},{1,21},{3,7},{1,23},{1,25},{1,27},{3,9},
a(14)=17={1,3},{1,5},{1,7},{1,9},{1,11},{1,13},{1,15},{3,5},{1,17},{1,19},{1,21},{3,7},{1,23},{1,25},{1,27},{3,9},{1,29},
a(15)=18={1,3},{1,5},{1,7},{1,9},{1,11},{1,13},{1,15},{3,5},{1,17},{1,19},{1,21},{3,7},{1,23},{1,25},{1,27},{3,9},{1,29},{1,31},
a(16)=20={1,3},{1,5},{1,7},{1,9},{1,11},{1,13},{1,15},{3,5},{1,17},{1,19},{1,21},{3,7},{1,23},{1,25},{1,27},{3,9},{1,29},{1,31},{1,33},{3,11},
a(17)=22={1,3},{1,5},{1,7},{1,9},{1,11},{1,13},{1,15},{3,5},{1,17},{1,19},{1,21},{3,7},{1,23},{1,25},{1,27},{3,9},{1,29},{1,31},{1,33},{3,11},{1,35},{5,7},
a(18)=23,

northwolves · 发表于 2025-1-17 16:27:50

王守恩发表于 2025-1-15 11:20
A020897——只计算到——120。

Table[FindInstance[{x^3 + y^3 == z^3 n, 0 < y, 0 < z < 2000}, {x, y, ...

Rational solutions of x^3+y^3=n

王守恩 · 发表于 2025-1-19 10:13:43

(1), A211264——整数对数{x, y},  使得 0 < x < y ≤ n 和 x*y ≤ n。

0, 1, 2, 3, 4, 6, 7, 9, 10, 12, 13, 16, 17, 19, 21, 23, 24, 27, 28, 31, 33, 35, 36, 40, 41, 43, 45, 48, 49, 53, 54, 57, 59, 61, 63, 67, 68, 70, 72, 76, 77, 81, 82, 85, 88, 90, 91, 96, 97, 100, 102, 105, 106, 110, 112, 116, 118, 120, 121,

Table[Sum[Floor[n/k] - k, {k, Sqrt[n]}], {n, 75}]

Table[Sum[Count[Flatten@Table[x*y, {x, n}, {y, x + 1, n}], k], {k, n}], {n, 75}]

(2), A181972——整数对数{x, y},  使得 0 < x < y ≤ n 和 x*y ≤ floor(n/2)。

0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 7, 7, 9, 9, 10, 10, 12, 12, 13, 13, 16, 16, 17, 17, 19, 19, 21, 21, 23, 23, 24, 24, 27, 27, 28, 28, 31, 31, 33, 33, 35, 35, 36, 36, 40, 40, 41, 41, 43, 43, 45, 45, 48, 48, 49, 49, 53, 53,

Table[Sum[Floor[n/(2 k)] - k, {k, Sqrt[n/2]}], {n, 60}]

Table[Sum[Count[Flatten@Table[x*y, {x, n/2}, {y, x + 1, n/2}], k], {k, n/2}], {n, 60}]

(3), OEIS 没有——1, 3, 6, 9, 12, 16, 19, 23, 27, 31, 35, 40, 43, 48, 53, 57, 61, 67, 70, 76, 81, 85, 90, 96, 100, 105, 110, 116, 120, 127, 130, 136, 142, 146, 152, 159, 162, 168, 174, 180, 184, 191,

a(2)=1={1,2},
a(4)=3={1,3},{1,4},
a(6)=6={1,5},{1,6},{2,3},
a(8)=9={1,7},{1,8},{2,4},
a(10)=12={1,9},{1,10},{2,5},
a(12)=16={1,11},{1,12},{2,6},{3,4},
a(14)=19={1,13},{1,14},{2,7},
a(16)=23={1,15},{3,5},{1,16},{2,8},

Table[Sum[Floor[2 n/k] - k, {k, Sqrt[2 n]}], {n, 75}]

(4), OEIS 没有————整数对数{x, y},  使得 1 < x < y ≤ n 和 x*y ≤ n。

0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 5, 5, 6, 7, 8, 8, 10, 10, 12, 13, 14, 14, 17, 17, 18, 19, 21, 21, 24, 24, 26, 27, 28, 29, 32, 32, 33, 34, 37, 37, 40, 40, 42, 44, 45, 45, 49, 49, 51, 52, 54, 54, 57, 58,

Table[Sum[Floor[n/k] - k, {k, 2, Sqrt[n]}], {n, 60}]

Table[Sum[Count[Flatten@Table[x*y, {x, 2, n/2}, {y, x + 1, n/2}], k], {k, n}], {n, 60}]

(5), 我们也没有？！——奇数对数{x, y},  使得 0 < x < y ≤ 2 n + 1 和 x*y ≤2 n + 1——这通项公式还不好找？！

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 22, 23, 25, 26, 27, 30, 31, 32, 34, 35, 37, 39, 40, 41, 44, 46, 47, 49, 50, 51, 54, 56, 57, 59, 60, 62, 64, 65, 67, 69, 71, 72, 74, 75,

a(0)=0
a(1)=1={1,3},
a(2)=2={1,5},
a(3)=3={1,7},
a(4)=4={1,9},
a(5)=5={1,11},
a(6)=6={1,13},
a(7)=8={1,15},{3,5},
a(8)=9={1,17},
a(9)=10={1,19},
a(10)=12={1,21},{3,7},
a(11)=13={1,23},
a(12)=14={1,25},
a(13)=16={1,27},{3,9},
a(14)=17={1,29},
a(15)=18={1,31},
a(16)=20={1,33},{3,11},
a(17)=22={1,35},{5,7},
a(18)=23,

王守恩 · 发表于 2025-1-20 08:59:07

(5), 我们也没有？！——奇数对数{x, y}, 使得 0 < x < y ≤ 2 n + 1 和 x*y ≤2 n + 1——这通项公式还不好找？！

昨天没有！今天可以有！！

{0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 22, 23, 25, 26, 27, 30, 31, 32, 34, 35, 37, 39, 40, 41, 44, 46, 47, 49, 50, 51, 54, 56, 57, 59, 60, 62, 64, 65, 67, 69, 71, 72, 75, 76, 77, 81,
82, 83, 85, 86, 88, 91, 93, 94, 96, 98, 99, 101, 102, 104, 108, 109, 110, 112, 114, 116, 119, 120, 121, 124, 126, 127, 129, 131, 132, 136, 137, 138, 141, 142, 145, 147, 148, 149, 151, 153, 155,
159, 160, 161, 165, 166, 167, 169, 171, 173, 176, 178, 179, 181, 183, 185, 187, 189, 190, 194, 195, 196, 200, 201, 203, 205, 206, 207, 210, 213, 215, 217, 218, 220, 224, 225, 227, 230, 231, ...

Table[Sum[Floor[(n - 2 k (k - 1))/(2 k - 1)], {k, Sqrt[(n + 1)/2]}], {n, 0, 60}]

Table[Sum[Floor[(n + k)/(2 k - 1)] - k, {k, Sqrt[(n + 1)/2]}], {n, 0, 60}]

Table[Sum[Floor[(n - 1 - 2 k (k - 1))/(2 k - 1)], {k, Sqrt[n/2]}], {n, 61}]

Table[Sum[Floor[(n - 1 + k)/(2 k - 1)] - k, {k, Sqrt[n/2]}], {n, 61}]

扩散一下——{0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 9, 9, 10, 11, 11, 11, 13, 14, 14, 15, 15, 15, 17, 18, 18, 19, 19, 20, 21, 21, 22, 23, 24, 24, 26, 26, 26, 29, 29, 29, 30, 30, 31, 33, 34, 34}
a(1)=0,
a(2)=0,
a(3)=0,
a(4)=0,
a(5)=0,
a(6)=0,
a(7)=0,
a(8)=1={3,5},
a(9)=1,
a(10)=1,
a(11)=2={3,7},
a(12)=2,
a(13)=2,
a(14)=3={3,9},
a(15)=3,
a(16)=3,
a(17)=4={3,11},
a(18)=5={5,7},
a(19)=5,

Table[Sum[Floor[(n + k - 1)/(2 k - 1)] - k, {k, 2, Sqrt[n/2]}], {n, 60}]

附带的把A339183也化简了。

A339183——0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6,

Table[Floor[Sqrt[(n - 1)/2]], {n, 60}]

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[原创] 数字串的通项公式

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