用 1，2，+，×，( )=正整数。

王守恩

用 1，2，+，×，( )。譬如：
a(1)=1,{1},1,
a(2)=1,{2},1,
a(3)=2,{1+2},1+1,
a(4)=2,{2*2},{2+2}x
a(5)=3,{1+4},{2+3},
a(6)=3,{2*3},{2+4},
a(7)=4,{1+6},{2+5},{3+4},
a(8)=3,{2*4},1+2,
a(9)=4,{3*3},{1+8},
a(10)=4,{2*5},{2+8},
a(11)=5,{1+10},{2+9},{3+8},
a(12)=4,{3*4},{2*6},
a(13)=5,{1+12},1+4,
a(14)=5,{2*7},{2+12},
a(15)=5,{3*5},2+3,
a(16)=4,{2*8},1+3,
a(17)=5,{1+16},1+4,
a(18)=5,{3*6},
a(19)=6,{1+18},
a(20)=5,{2*10},1+4,
a(21)=6,{3*7},2+4
a(22)=6,{2*11},1+5,
a(23)=7,{2+21},
a(24)=5,{3*8},2+3,
a(25)=6,{1+24},1+5,
a(26)=6,{2*13},
a(27)=6,{3*9},2+4,
a(28)=6,{2*14},1+5,
a(29)=7,{2+27},
a(30)=6,{3*10},1+5,

第1个数 = 1, 2, 3, 5, 7, 11, 13, 17, 19, ...
与上次《用 2，3，+，×，( )=正整数》第1个数 = 2, 3, 5, 7, 11, 13, 17, 19, ...
这次还困难些。多了个不好对付的 ”1“。

nyy

你这个问题有什么意义？我只好奇背后的意义，我对这个问题不感兴趣

northwolves

第1个数 = 3, 5, 7, 11, 13, 17, 19,...显然是p^2

王守恩

1 楼是这样一串数——1, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 5, 4, 5, 5, 6, 5, 6, 6, 7, 5, 6, 6, 6, 6, 7, 6, 7, 5, 7, 6, 7, 6, 7, 7, 7, 6, 7, 7, 8, 7, 7, 8, 8, 6, 8, 7, 7, 7, 8, 7, 8, 7, 8, 8, 9, 7, 8, 8, 8, 6, 8, 8, 9, 7, 9, 8, 9, 7, 8, 8, 8, 8, 9, 8, 9, 7, 8,  

A064097相近的数——0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 5, 4, 5, 5, 6, 5, 6, 6, 7, 5, 6, 6, 6, 6, 7, 6, 7, 5, 7, 6, 7, 6, 7, 7, 7, 6, 7, 7, 8, 7, 7, 8, 9, 6, 8, 7, 7, 7, 8, 7, 8, 7, 8, 8, 9, 7, 8, 8, 8, 6, 8, 8, 9, 7, 9, 8, 9, 7, 8, 8, 8, 8, 9, 8, 9, 7, 8,

1楼的数(手工计算会有误差)肯定不是A064097——A064097会出现7,8,9, ... 8,9,10, ... 9,10,11, ...。而1楼的数肯定不会出现7,8,9, ... 8,9,10, ... 9,10,11, ...。

王守恩

northwolves 发表于 2025-5-27 21:23
第1个数 = 3, 5, 7, 11, 13, 17, 19,...显然是p^2

要不这样想。(2)=(1)+(1)=(1)*(1),   (3)=(1)+(2)=(1)*(2),   (4)=(1)+(3)=(2)+(2)=(1)*(3)=(2)*(2),  ......有规律吗？
(1) = 1, 2,
(2) = 3, 4,
(3) = 5, 6, 8,
(4) = 7, 9, 10, 12, 13, 16,
(5) = 11, 13, 14, 15, 17, 18, 20, 24, 32,
(6) = 19, 21, 22, 25, 26, 27, 28, 30, 33, 34, 36, 40, 48, 64,
(7) = 23, 29, 31, 35, 37, 38, 39, 41, 42, 44, 45, 49, 50, 51, 52, 54, 56, 60, 65, 66, 68, 72, 80,
(8) = 43, 46, 47, 53, 55, 57, 58, 61, 62, 63, 67, 69, 70, 73, 74, 75, 76, 77, 78, 81, 62, 84, 85, 66, 90,
(9) = 59, 71, 79, 83, 86, 87, 89,
(10)=107,
(11)=173,
(12)=283,

{1, 3, 5, 7, 11, 19, 23, 43, 59, 107, 173, 283, 383, 719, 1103, 1439, 3019, 4283, 8563, 14207}——我还是搞不出来！  

northwolves

王守恩 发表于 2025-5-28 06:31
要不这样想。(2)=(1)+(1)=(1)*(1),   (3)=(1)+(2)=(1)*(2),   (4)=(1)+(3)=(2)+(2)=(1)*(3)=(2)*(2),  .. ...

1. num=1500;a=Array[20&,num];
   
2. a[[1]]=a[[2]]=1;
   
3. Do[Do[s=a[[k]]+a[[n-k]];If[s<a[[n]],a[[n]]=s];If[Mod[n,k]==0,s=a[[k]]+a[[n/k]];If[s<a[[n]],a[[n]]=s]],{k,n/2}],{n,3,num}];
   
4. Table[FirstPosition[a,k][[1]],{k,16}]

复制代码

{1,3,5,7,11,19,23,43,59,107,173,283,383,719,1103,1439}

王守恩

northwolves 发表于 2025-5-29 16:23
{1,3,5,7,11,19,23,43,59,107,173,283,383,719,1103,1439}

这串数也来几个——1楼——蛮实用的一串数——OEIS没有这串数——我被搞晕了！谢谢！

northwolves

{1, 3, 5, 7, 11, 19, 23, 43, 59, 107, 173, 283, 383, 719, 1103, 1439, 3019, 4283, 8563, 14207, 40013, 41399, 80437 }

{1,1,1}
{3,(1+2),2}
{5,(1+(2+2)),3}
{7,(1+(2+(2+2))),4}
{11,(1+(2+2*(2+2))),5}
{19,(1+(2+2*2*(2+2))),6}
{23,(1+(2+2*(2+2*(2+2)))),7}
{43,(1+(2+2*2*(2+2*(2+2)))),8}
{59,(1+(2+2*2*(2+2*(2+(2+2))))),9}
{107,(1+(2+2*2*(2+2*2*(2+(2+2))))),10}
{173,(1+(2+2*(1+(2+2))*(1+2*2*(2+2)))),11}
{283,(1+(2+2*2*(2+2*(2+2*2*2*(2+2))))),12}
{383,(1+(2+2*2*(1+(2+2))*(1+(2+2*2*(2+2))))),13}
{719,(1+2*(2+(1+2)*(1+(2+(2+2)))*(1+2*2*(2+2)))),14}
{1103,(1+(2+2*(2+2*(2+2*2*2*(2+2*2*2*(2+2)))))),15}
{1439,(1+(2+2*2*(2+(1+2)*(1+(2+(2+2)))*(1+2*2*(2+2))))),16}
{3019,(1+(2+2*2*(2+2*2*2*2*(2+(1+2)*(1+2)*(1+(2+2)))))),17}
{4283,(1+(2+2*2*(2+2*(2+2*(2+2*2*(2+2*2*2*2*(2+2))))))),18}
{8563,(1+(2+2*2*2*(2+2*(2+2*(2+2*2*(2+2*2*2*2*(2+2))))))),19}
{14207,(1+(2+2*2*(2+(1+2*(2+(2+2)))*(1+2*2*2*(2+2*2*2*(2+2)))))),20}
{40013,(1+(2+2*(1+(2+2))*(1+2*2*2*2*2*(1+(2+2))*(1+2*2*(2+(2+2)))))),21}
{41399,(1+(2+2*(2+2*2*2*(2+(1+(2+2))*(1+2*(2+2*2*2*2*2*2*(2+2))))))),22}
{80437,(1+(2+2*(1+(2+2*2*2*2*2*(2+2)))*(1+2*(1+2)*(1+2)*(1+2*2*(2+2))))),23}

王守恩

northwolves 发表于 2025-5-29 21:21
{1, 3, 5, 7, 11, 19, 23, 43, 59, 107, 173, 283, 383, 719, 1103, 1439, 3019, 4283, 8563, 14207, 40013 ...

漂亮！让人遐想！！OEIS有吗！！！

a(1)=1, 1,
a(2)=3, 1+2,
a(3)=5, 1+2×2,
a(4)=7, 1+2+2×2,
a(5)=11, 1+2+2×2×2,
a(6)=19, 1+2+2×2×2×2,
a(7)=23, 1+2+2(2+2×2×2),
a(8)=43, 1+2+2×2(2+2×2×2),
a(9)=59, 1+2+2×2(2+2(2+2×2)),
a(10)=107, 1+2+2×2(2+2×2(2+2×2)),
a(11)=173, 1+2+2(1+2×2)(1+2×2×2×2),
a(12)=283, 1+2+2×2(2+2(2+2×2×2×2×2)),
a(13)=383, 1+2+2×2(1+2×2)(1+(2+2×2×2×2)),
a(14)=719, 1+2(2+(1+2)(1+2+2×2)(1+2×2×2×2)),
a(15)=1103, 1+2+2(2+2(2+2×2×2(2+2×2×2×2×2))),
a(16)=1439, 1+2+2×2(2+(1+2)(1+2+2×2)(1+2×2×2×2)),
a(17)=3019, 1+2+2×2(2+2×2×2×2(2+(1+2)(1+2)(1+2×2))),
a(18)=4283, 1+2+2×2(2+2(2+2(2+2×2(2+2×2×2×2×2×2)))),
a(19)=8563, 1+2+2×2×2(2+2(2+2(2+2×2(2+2×2×2×2×2×2)))),
a(20)=14207, 1+2+2×2(2+(1+2(2+2×2))(1+2×2×2(2+2×2×2×2×2))),
a(21)=40013, 1+2+2(1+2×2)(1+2×2×2×2×2(1+2×2)(1+2×2(2+2×2))),
a(22)=41399, 1+2+2(2+2×2×2(2+(1+2×2)(1+2(2+2×2×2×2×2×2×2×2)))),
a(23)=80437, 1+2+2(1+2(1+2)(1+2)(1+2×2×2×2))(1+(2+2×2×2×2×2×2×2)),

贪心一点：每个算式是唯一的吗？

northwolves

肯定不是唯一的啦   2+2=2*2