数字串的通项公式

王守恩

409#后半部分。
将3,4,5,6,…,n排成数列{a(k):k=1,2,3,4,5,…,n-2},使a(k)都是k的倍数,有几种排法?
a(01)=b(03)=1,{3},
a(02)=b(03)*b(04)=1,{3,4},
a(03)=b(04)*b(05)=1,{5,4,3},
a(04)=b(05)*b(06)=1,{5,6,3,4},
a(05)=b(06)*b(07)=1,{7,6,3,4,5},
a(06)=b(07)*b(08)=2,{7,4,3,8,5,6},{7,8,3,4,5,6},
a(07)=b(08)*b(09)=4,{3,4,8,9,5,6,7},{3,8,9,4,5,6,7},{9,4,3,8,5,6,7}{9,8,3,4,5,6,7},
a(08)=b(09)*b(10)=2,{3,10,9,4,5,6,7,8},{9,10,3,4,5,6,7,8},
a(09)=b(10)*b(11)=1,{11,10,3,4,5,6,7,8,9},
a(10)=b(11)*b(12)=3,{11,4,3,12,5,6,7,8,9,10},{11,6,3,4,5,12,7,8,9,10},{11,12,3,4,5,6,7,8,9,10},
a(11)=b(12)*b(13)=3,{13,4,3,12,5,6,7,8,9,10,11},{13,6,3,4,5,12,7,8,9,10,11},{13,12,3,4,5,6,7,8,9,10,11},
a(12)=b(13)*b(14)=1,{13,14,3,4,5,6,7,8,9,10,11,12},
a(13)=b(14)*b(15)=3,
得到这样一串数:1,1,1,1,1,2,4,2,1,3,3,1,3,12,4,2,2,3,9,3,1,8,16,2,4,12,3,3,3,8,8,...
b(03)=01,{3},   (奇数约数, 除了"1")
b(04)=01,{4},   (偶数约数, 除了"2")
b(05)=01,{5},   (奇数约数, 除了"1")
b(06)=01,{6},   (偶数约数, 除了"2")
b(07)=01,{7},          .........
b(08)=02,{4,8},{8},
b(09)=02,{3,9},{9},
b(10)=01,{10},
b(11)=01,{11},
b(12)=03,{4,12},{6,12},{12},
b(13)=01,{13},
b(14)=01,{14},
b(15)=03,{3,15},{5,15},{15},
b(16)=04,{4,8,16},{4,16},{8,16},{16}
b(17)=01,{17},
b(18)=02,{6,18},{18},
b(19)=01,{19},
b(20)=03,{4,20},{10,20},{20},
b(21)=03,{3,21},{7,21},{21},
b(22)=01,{22},
b(23)=01,{23},
b(24)=08,{4,8,24},{4,12,24},{4,24},{6,12,24},{6,24},{8,24},{12,24},{24},
b(25)=02,{5,25},{25},
b(26)=01,{26},
b(27)=04,{3,9,27},{3,27},{9,27},{27},
b(28)=03,{4,28},{14,28},{28},
b(29)=01,{29},
b(30)=03,{6,30},{10,30},{30},
b(31)=01,{31},
b(32)=08,{4,8,16,32},{4,8,32},{4,16,32},{4,32},{8,16,32},{8,32},{16,32},{32},
b(33)=01,{3,33},{11,33},{33},
b(n)可以有通项公式吗？谢谢各位！

王守恩

接411#。再归纳，各位可有好办法？
b(03)=1,{3},   
b(04)=1,{4},   
b(05)=1,{5},   
b(06)=1,{6},   
b(07)=1,{7},         
b(08)=2,{4,8},{8},
b(09)=2,{3,9},{9},
b(10)=1,{10},
b(11)=1,{11},
b(12)=3,{4,12},{6,12},{12},
b(13)=1,{13},
b(14)=1,{14},
b(15)=3,{3,15},{5,15},{15},
b(16)=4,{4,8,16},{4,16},{8,16},{16}
b(17)=1,{17},
b(18)=2,{6,18},{18},
b(19)=1,{19},
b(20)=3,{4,20},{10,20},{20},
......
01=3,5,07,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,
01=4,6,10,14,22,26,34,38,46,58,62,74,82,86,94,
02=9,25,49,121,
02=8,18,50,098,
03=15,21,33,35,39,51,55,57,65,69,77,85,87,93,95,
03=12,20,28,30,42,44,52,66,68,70,76,78,92,
04=27,125,
04=16,054,
08=45,63,75,81,99,
08=24,32,36,40,56,88,
09=90,
13=60,
16=64,
20=48,080,
26=72,200,

northwolves

01   奇素数
02 奇素数平方
03 两不同奇素数之积

王守恩

将2,3,4,5,6,…,n排成数列{a(k):k=1,2,3,4,5,…,n-1},使a(k)都是k的倍数,有几种排法?

将3,4,5,6,…,n排成数列{a(k):k=1,2,3,4,5,…,n-2},使a(k)都是k的倍数,有几种排法?

将4,5,6,…,n排成数列{a(k):k=1,2,3,4,5,…,n-3},使a(k)都是k的倍数,有几种排法?

northwolves

王守恩 发表于 2023-10-25 06:01
将2,3,4,5,6,…,n排成数列{a(k):k=1,2,3,4,5,…,n-1},使a(k)都是k的倍数,有几种排法?

将3,4,5,6,…,n排成 ...

1. a[n_,k_]:=Length@Select[Permutations[Range[k,n+k-1],{n}],Sum[Mod[#[[i]],i],{i,n}]==0&];Table[a[n,k],{k,2,4},{n,9}]

复制代码

{{1,1,2,1,3,1,4,2,3},{1,1,1,1,1,2,4,2,1},{1,1,1,0,2,2,6,1,2}}

王守恩

将4,5,6,…,2024排成数列{a(k):k=1,2,3,4,5,…,2021},使a(k)都是k的倍数,有几种排法?

王守恩

将4,5,6,…,2024排成数列{a(k):k=1,2,3,4,5,…,2021},使a(k)都是k的倍数,有几种排法?
答: 有8(2023)*44(2024)*3(2022)=1056种排法。
a(1)=2023,
01: a(1)=7,a(7)=119,a(119)=2023,
02: a(1)=7,a(7)=2023,
03: a(1)=17,a(17)=119,a(119)=2023,
04: a(1)=17,a(17)=289,a(289)=2023,
05: a(1)=17,a(17)=2023,
06: a(1)=119,a(119)=2023,
07: a(1)=289,a(289)=2023,
08: a(1)=2023,
a(2)=2024,
01: a(2)=4,a(4)=8,a(8)=88,a(88)=2024,
02: a(2)=4,a(4)=8,a(8)=184,a(184)=2024,
03: a(2)=4,a(4)=8,a(8)=2024,
04: a(2)=4,a(4)=44,a(44)=88,a(88)=2024,
05: a(2)=4,a(4)=44,a(44)=1012,a(1012)=2024,
06: a(2)=4,a(4)=44,a(44)=2024,
07: a(2)=4,a(4)=88,a(88)=2024,
08: a(2)=4,a(4)=92,a(92)=184,a(184)=2024,
09: a(2)=4,a(4)=92,a(92)=1012,a(1012)=2024,
10: a(2)=4,a(4)=92,a(92)=2024,
11: a(2)=4,a(4)=184,a(184)=2024,
12: a(2)=4,a(4)=1012,a(1012)=2024,
13: a(2)=4,a(4)=2024,
14: a(2)=8,a(8)=88,a(88)=2024,
15: a(2)=8,a(8)=184,a(184)=2024,
16: a(2)=8,a(8)=2024,
17: a(2)=22,a(22)=44,a(44)=88,a(88)=2024,
18: a(2)=22,a(22)=44,a(44)=1012,,a(1012)=2024,
19: a(2)=22,a(22)=44,a(44)=2024,
20: a(2)=22,a(22)=88,a(88)=2024,
21: a(2)=22,a(22)=506,a(506)=1012,a(1012)=2024,
22: a(2)=22,a(22)=506,a(506)=2024,
23: a(2)=22,a(22)=1012,a(1012)=2024,
24: a(2)=22,a(22)=2024,
25: a(2)=44,a(44)=88,a(88)=2024,
26: a(2)=44,a(44)=1012,a(1012)=2024,
27: a(2)=44,a(44)=2024,
28: a(2)=46,a(46)=92,a(92)=184,a(184,)=1012,a(1012)=2024,
29: a(2)=46,a(46)=92,a(92)=1012,a(1012)=2024,
30: a(2)=46,a(46)=92,a(92)=2024,
31: a(2)=46,a(46)=184,a(184)==2024,
32: a(2)=46,a(46)=506,a(506)=1012,a(1012)==2024,
33: a(2)=46,a(46)=506,a(506)=2024,
34: a(2)=46,a(46)=1012,a(1012)=2024,
35: a(2)=46,a(46)=2024,
36: a(2)=88,a(88)=2024,
37: a(2)=92,a(92)=184,a(184)=2024
38: a(2)=92,a(92)=1012,a(1012)=2024,
39: a(2)=92,a(92)=2024,
40: a(2)=184,a(184)=2024,
41: a(2)=506,a(506)=1012,a(1012)=2024,
42: a(2)=506,a(506)=2024,
43: a(2)=1012,a(1012)=2024,
44: a(2)=2024,
a(3)=2022,
01: a(3)=6,a(6)=2022,
02: a(3)=1011,a(1011)=2022,
03: a(3)=2022,

王守恩

417#简化。将4,5,6,…,2024排成数列{a(k):k=1,2,3,4,5,…,2021},使a(k)都是k的倍数,有几种排法?
答: 有8*44*3=1056种排法。
a(1)=8,
01: 7,119,2023,
02: 7,2023,
03: 17,119,2023,
04: 17,289,2023,
05: 17,2023,
06: 119,2023,
07: 289,2023,
08: 2023,
a(2)=44,
01: 4,8,88,2024,
02: 4,8,184,2024,
03: 4,8,2024,
04: 4,44,88,2024,
05: 4,44,1012,2024,
06: 4,44,2024,
07: 4,88,2024,
08: 4,92,184,2024,
09: 4,92,1012,2024,
10: 4,92,2024,
11: 4,184,2024,
12: 4,1012,2024,
13: 4,2024,
14: 8,88,2024,
15: 8,184,2024,
16: 8,2024,
17: 22,44,88,2024,
18: 22,44,1012,2024,
19: 22,44,2024,
20: 22,88,2024,
21: 22,506,1012,2024,
22: 22,506,2024,
23: 22,1012,2024,
24: 22,2024,
25: 44,88,2024,
26: 44,1012,2024,
27: 44,2024,
28: 46,92,184,1012,2024,
29: 46,92,1012,2024,
30: 46,92,2024,
31: 46,184,2024,
32: 46,506,1012,2024,
33: 46,506,2024,
34: 46,1012,2024,
35: 46,2024,
36: 88,2024,
37: 92,184,2024,
38: 92,1012,2024,
39: 92,2024,
40: 184,2024,
41: 506,1012,2024,
42: 506,2024,
43: 1012,2024,
44: 2024,
a(3)=3,
01: 6,2022,
02: 1011,2022,
03: 2022,

王守恩

A138588       
       
2, 3, 4, 6, 6, 20, 24, 48, 48, 110, 110, 110, 243, 403, 402, 2504, 2352, 12219, 25200, 60458, 14256, 95760,

a(1)=02, 02(1),  2是1的倍数。
a(2)=03, 03(1),04(2),  3是1的倍数,4是2的倍数。
a(3)=04, 04(2),05(1),06(3),  4是2的倍数,5是1的倍数,6是3的倍数。
a(4)=06, 06(2),07(1),08(4),09(3),
a(5)=06, 06(2),07(1),08(4),09(3),10(5),
a(6)=20, 20(4),21(3),22(2),23(1),24(6),25(5),
a(7)=24, 24(4),25(5),26(2),27(3),28(7),29(1),30(6),
a(8)=48, 48(8),49(7),50(2),51(3),52(4),53(1),54(6),55(5),
a(9)=48, 48(6),49(7),50(2),51(3),52(4),53(1),54(9),55(5),56(8),

各位网友！可有通项公式？谢谢！

王守恩

将10,11,12,…,56排成数列{a(k):k=1,2,3,4,5,…,47},使a(k)都是k的倍数,有几种排法?
答: 有1*2*1*1*1*4*1*1*3=24种排法。
a(1)=1,
1: 53,
a(2)=2,
1: 10,50,
2: 50,
a(3)=1,
1: 51,
a(4)=1,
1: 52,
a(5)=1,
1: 55,
a(6)=4,
1: 12,24,
2: 12,48,
3: 24,48,
4: 48,
a(7)=1,
1: 49,
a(8)=1,
1: 56,
a(9)=3,
1: 18,54,
2: 27,54,
3: 54,