数字串的通项公式

northwolves

f[1012]=10；f[674]=10；f[2023]=12；f[2024]=TerminatedEvaluation["RecursionLimit"]

注意到 2024=1012*2， f[2024]=f[1012]+1=11

northwolves

改进并简化算法：

1. f[n_] :=
   
2.   f[n] = If[n == 0, 0,
   
3.     1 + Min[Mod[n, 2] + f[Floor[n/2]], Mod[n, 3] + f[Floor[n/3]]]];

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1. f[4478975]
   
2. f[16796159]

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34
36

王守恩

A085315——才40项——这里有88项——通项公式也没有我们的好！！！

{1, 2, 7, 11, 101, 111, 1001, 1011, 1101, 10001, 10011, 10101, 11001, 11011, 100001, 100011, 100101, 100111, 101001, 101011, 101101, 110001, 110011, 110101, 111001, 1000001,
1000011, 1000101, 1000111, 1001001,  1001011, 1001101, 1010001, 1010011, 1011001, 1100001, 1100011, 1100101, 1101001, 1110001, 10000001,  10000011, 10000101, 10000111,
10001001, 10001011, 10001101, 10010001, 10010011, 10010101, 10011001, 10100001,10100011, 10100101,10101001, 10110001, 11000001,11000011,11000101,11001001,11010001,
11100001, 100000001, 100000011, 100000101, 100000111, 100001001, 100001011, 100001101, 100010001, 100010011, 100011001, 100100001, 100100011, 100100101,  100101001,
100110001, 101000001, 101000011, 101000101, 101001001, 101100001, 110000001, 110000011, 110000101, 110001001, 110010001, 110100001, 111000001}

1. Select[Range[(10^9 - 1)/9], IntegerReverse[#]^3 == IntegerReverse[#^3] && Mod[#, 10] != 0 &]

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王守恩

northwolves 发表于 2025-2-16 12:27
改进并简化算法：

Solve[{IntegerReverse[x^3] == y^3, y ≥ x, 9000 > x > 0}, {x, y}, Integers]

{{x -> 1, y -> 1}, {x -> 2, y -> 2}, {x -> 7, y -> 7}, {x -> 11,  y -> 11}, {x -> 101, y -> 101}, {x -> 111, y -> 111}, {x -> 1001, y -> 1001}, {x -> 1011, y -> 1101}, {x -> 2201, y -> 2201}}

我们只想把 x 拉出来——1, 2, 7, 11, 101, 111, 1001, 1011, 2201,......我这里出不来了。

就这几个数——1, 2, 7, 11, 101, 111, 1001, 1011, 2201——OEIS——没有这串数。

northwolves

王守恩 发表于 2025-2-18 08:42
Solve[{IntegerReverse[x^3] == y^3, y ≥ x, 9000 > x > 0}, {x, y}, Integers]

{{x -> 1, y -> 1}, {x ...

1. Select[Range@1000000, Mod[#, 10] > 0 && IntegerQ@CubeRoot@IntegerReverse[#^3] &]

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{1, 2, 7, 11, 101, 111, 1001, 1011, 1101, 2201, 10001, 10011, 10101, 11001, 11011, 100001, 100011, 100101, 100111, 101001, 101011, 101101,110001, 110011, 110101, 111001}

northwolves

1. Select[Range@1000000,IntegerReverse@#>=#&&IntegerQ@CubeRoot@IntegerReverse[#^3]&]

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王守恩

看另外一串数——OEIS也没有。
Solve[{IntegerReverse[x^2] == y^2, y >= x, 9000 > x > 0}, {x, y}, Integers]
Flatten@Solve[{IntegerReverse[x^2] == y^2, y >= x, 9000 > x > 0}, {x, y}, Integers]
{x -> 1, y -> 1, x -> 2, y -> 2, x -> 3, y -> 3, x -> 11, y -> 11, x -> 12, y -> 21, x -> 13, y -> 31, x -> 22, y -> 22, x -> 26, y -> 26, x -> 33, y -> 99, x -> 101, y -> 101, x -> 102, y -> 201, x -> 103, y -> 301,
x -> 111, y -> 111, x -> 112, y -> 211, x -> 113, y -> 311, x -> 121, y -> 121, x -> 122, y -> 221, x -> 202, y -> 202, x -> 212, y -> 212, x -> 264, y -> 264, x -> 307, y -> 307, x -> 836, y -> 836, x -> 1001, y -> 1001,
x -> 1002, y -> 2001, x -> 1003, y -> 3001, x -> 1011, y -> 1101, x -> 1012, y -> 2101, x -> 1013, y -> 3101, x -> 1021, y -> 1201, x -> 1022, y -> 2201, x -> 1031, y -> 1301, x -> 1102, y -> 2011, x -> 1103, y -> 3011,

只要 x ——1, 2, 3, 11, 12, 13, 22, 26, 33, 101, 102, 103, 111, 112, 113, 121, 122, 202, 212, 264, 307, 836, 1001, 1002, 1003, 1011, 1012, 1013, 1021, 1022, 1031, 1102, 1103,

northwolves

1. Select[Range@2000, IntegerReverse[#^2] >= #^2 && IntegerQ@Sqrt@IntegerReverse[#^2] &]

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{1,2,3,11,12,13,22,26,33,101,102,103,111,112,113,121,122,202,212,264,307,836,1001,1002,1003,1011,1012,1013,1021,1022,1031,1102,1103,1111,1112,1113,1121,1122,1202,1212}

northwolves

王守恩 发表于 2025-2-18 08:42
Solve[{IntegerReverse[x^3] == y^3, y ≥ x, 9000 > x > 0}, {x, y}, Integers]

{{x -> 1, y -> 1}, {x ...

1. Select[Range@100000,IntegerReverse[#^3]>=#^3&&IntegerQ@CubeRoot@IntegerReverse[#^3]&]
   
2. 
   

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{1, 2, 7, 11, 101, 111, 1001, 1011, 2201, 10001, 10011, 10101, 11011}

王守恩

Table[Select[Range@2000000, IntegerQ[Power[IntegerReverse[#^k], (1/k)]] && IntegerReverse[#^k] ≥ #^k &], {k, 4, 9}]

简单的捋一捋。题目。x^k = A,  y^k = B, 其中 A 是 B 的颠倒整数。当然也可以说 B 是 A 的颠倒整数。

x 与 y 是不可分割的一对数，找到 x,   y 也就自动出来了。不妨约定 y ≥ x。我们只要找 x 就可以了。OEIS有点乱。

k = 5, 6, 7, 8, 9, ......—— 没有解。

k = 4—— 只有1个解——{1, 11, 101, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001, 10000000001, 100000000001,

k = 3 ——后面不会出现2——{1, 2, 7, 11, 101, 111, 1001, 1011, 2201, 10001, 10011, 10101, 11011, 100001, 100011, 100101, 100111, 101011, 101101, 110011, 1000001, 1000011, 1000101, 1000111, 1001001,
1001011, 1001101, 1010011, 1100011, 10000001, 10000011, 10000101, 10000111, 10001001, 10001011, 10001101, 10010011, 10010101, 10011001, 10100011, 10100101, 11000011,

k = 2 就这个复杂一点——{1, 2, 3, 11, 12, 13, 22, 26, 33,  101, 102, 103, 111, 112, 113, 121, 122, 202, 212, 264, 307, 836,  1001, 1002, 1003, 1011, 1012, 1013, 1021, 1022, 1031, 1102, 1103, 1111, 1112, 1113,
1121, 1122, 1202, 1212, 2002, 2012, 2022, 2285, 2636, 3168, 10001, 10002, 10003, 10011, 10012, 10013, 10021, 10022, 10031, 10101, 10102, 10103, 10111, 10112, 10113, 10121, 10122, 10201, 10202, 10211,
10212, 10221, 11002, 11003, 11011, 11012, 11013, 11021,  11022, 11031, 11102, 11103, 11111, 11112, 11113,  11121, 11122, 11202, 11211, 12002, 12012, 12102, 12202, 20002, 20012,  20022, 20102, 20112,
20122, 20508, 22865, 24846, 30693, 100001, 100002, 100003, 100011, 100012, 100013, 100021, 100022, 100031, 100101,100102, 100103, 100111, 100112, 100113, 100121, 100122, 100201, 100202, 100211,
100212, 100221, 100301, 100311, 101002, 101003, 101011, 101012, 101013, 101021, 101022, 101031, 101101, 101102, 101103, 101111, 101112, 101113, 101121, 101122, 101201, 101202, 101211, 101212,
101301, 102002, 102011, 102012, 102021, 102022, 102102, 102111, 102121, 110002, 110003, 110011, 110012, 110013, 110021, 110022, 110031, 110102, 110103, 110111, 110112, 110113, 110121, 110122,
110202, 110211, 110212, 110221, 110922, 111002, 111003, 111012, 111013, 111021, 111022, 111031, 111102, 111103, 111111, 111112, 111121, 111202, 111211, 112002, 112012, 112102, 120002, 120012,
120102, 120112, 121002, 121102, 122002, 200002, 200012, 200022, 200102, 200112, 200122, 200202, 200212, 201012, 201022, 202012, 303577, 798644,

OEIS没有k = 3,    OEIS没有k = 2。这2个数字串是你的！我是不管了。

又: k = 3有点慢, 我们就规定 x 是1与0组成的十进制数,  会快一些,  我编不了。

再开个玩笑——每个数有>3数码的——拉出来。我是拉不出来的。
26, 264, 307, 836, 2285, 2636, 3168, 20508, 22865, 24846, 30693, 110922, 303577, 798644, 1042151, 1100922, 1109111, 1109211, 1110922, 1270869, 2012748, 2294675, 3069307, 3080367, 10110922, 11009111, 11009122,
11009221, 11091022, 11091111, 11091121, 11091122, 11091202, 11091211, 11091212, 11092111, 11100922, 11109211, 11110922, 11129361, 12028229, 12866669, 26049013, 30001253, 31955891, 64030648}