OEIS序列A325112的通项公式

王守恩

《整数序列在线百科全书(OEIS)》A325112  整数，使得十进制表示的非零子序列不能被 3 整除。

1, 2, 4, 5, 7, 8, 10, 11, 14, 17, 20, 22, 25, 28, 40, 41, 44, 47, 50, 52, 55, 58, 70, 71, 74, 77, 80, 82, 85, 88,

可以有通项公式吗？

王守恩

《整数序列在线百科全书(OEIS)》A325112  整数，使得十进制表示的非零子序列不能被 3 整除。

1, 2, 4, 5, 7, 8, 10, 11, 14, 17, 20, 22, 25, 28, 40, 41, 44, 47, 50, 52, 55, 58, 70, 71, 74, 77, 80, 82, 85, 88, 100,
101, 104, 107, 110, 140, 170, 200, 202, 205, 208, 220, 250, 280, 400, 401, 404, 407, 410, 440, 470, 500, 502, 505,
508, 520, 550, 580, 700, 701, 704, 707, 710, 740, 770, 800, 802, 805, 808, 820, 850, 880, 1000,1001, 1004, ......

细腻的通项公式没有，粗糙的通项公式可以有。

a(1)=1
a(7)=10
a(31)=100
a(73)=1000
a(133)=10000
a(211)=100000
a(307)=1000000
a(421)=10000000
a(553)=100000000

王守恩

  正整数中: 使得十进制表示的非零子序列不能被 3 整除。

{1, 2, 4, 5, 7, 8, 10, 11, 14, 17, 20, 22, 25, 28, 40, 41, 44, 47, 50, 52, 55, 58, 70, 71, 74, 77, 80, 82, 85, 88, 100, 101, 104, 107,
110, 140, 170, 200, 202, 205, 208, 220, 250, 280, 400, 401, 404, 407, 410, 440, 470, 500, 502, 505, 508, 520, 550, 580, 700}

OEIS--A325112给出公式。

With[{k = 3}, Select[Range@700, NoneTrue[DeleteCases[FromDigits /@Rest@Subsequences[IntegerDigits@#], 0], Mod[#, k] == 0 &] &]]

我们可不可以有这样一个算式(可我不会)：取这些数的倒数，然后相加，和是多少？
我的理想：
我们有第一个按钮，把这串数从小到大排列起来：a(1)=1,a(2)=2,a(3)=4,a(4)=5,a(5)=7,....
我们有第二个按钮，把这串数累加起来：s=1/a(1)+1/a(2)+1/a(3)+a(4)+1/a(5)+....

northwolves

王守恩 发表于 2022-7-15 07:32
《整数序列在线百科全书(OEIS)》A325112  整数，使得十进制表示的非零子序列不能被 3 整除。

1, 2, 4,  ...

$a(9 n^2-3 n+1)=10^n$

northwolves

$a(9 n^2+3kn-2k-1)=[\sqrt{2}k]*10^n,k \in [1,6]$

northwolves

算了半小时才凑对
$m=3*\floor({\sqrt{4n-3}+1)/6)+1$
$x=\floor{\frac{n-3}{m}}-(m-3)$
$y=(n-3)%m-1$
$z=\floor(1+(3*(x%6))/2)$
$a_n=z*10^{\floor({\sqrt{4n-3}+1)/6)}+if(y<0,0,(3*(y%3)+z%3)*10^{\floor{\frac{y}{3}}})$

northwolves

1. n=200
   
2. s=[]
   
3. for i in range(1,n+1):
   
4.     m=3*int(((4*i-3)**0.5+1)/6)+1
   
5.     x,y=(i-3)//m-(m-3),(i-3)%m-1
   
6.     z=int(1+(3*(x%6))/2)
   
7.     w=z*10**(m//3)
   
8.     if y>-1: w+=(3*(y%3)+z%3)*10**(y//3)
   
9.     s.append(w)
   
10. print(s)

复制代码

[1, 2, 4, 5, 7, 8, 10, 11, 14, 17, 20, 22, 25, 28, 40, 41, 44, 47, 50, 52, 55, 58, 70, 71, 74, 77, 80, 82, 85, 88, 100, 101, 104, 107, 110, 140, 170, 200, 202, 205, 208, 220, 250, 280, 400, 401, 404, 407, 410, 440, 470, 500, 502, 505, 508, 520, 550, 580, 700, 701, 704, 707, 710, 740, 770, 800, 802, 805, 808, 820, 850, 880, 1000, 1001, 1004, 1007, 1010, 1040, 1070, 1100, 1400, 1700, 2000, 2002, 2005, 2008, 2020, 2050, 2080, 2200, 2500, 2800, 4000, 4001, 4004, 4007, 4010, 4040, 4070, 4100, 4400, 4700, 5000, 5002, 5005, 5008, 5020, 5050, 5080, 5200, 5500, 5800, 7000, 7001, 7004, 7007, 7010, 7040, 7070, 7100, 7400, 7700, 8000, 8002, 8005, 8008, 8020, 8050, 8080, 8200, 8500, 8800, 10000, 10001, 10004, 10007, 10010, 10040, 10070, 10100, 10400, 10700, 11000, 14000, 17000, 20000, 20002, 20005, 20008, 20020, 20050, 20080, 20200, 20500, 20800, 22000, 25000, 28000, 40000, 40001, 40004, 40007, 40010, 40040, 40070, 40100, 40400, 40700, 41000, 44000, 47000, 50000, 50002, 50005, 50008, 50020, 50050, 50080, 50200, 50500, 50800, 52000, 55000, 58000, 70000, 70001, 70004, 70007, 70010, 70040, 70070, 70100, 70400, 70700, 71000, 74000, 77000, 80000, 80002, 80005]

王守恩

northwolves 发表于 2022-12-30 23:01
[1, 2, 4, 5, 7, 8, 10, 11, 14, 17, 20, 22, 25, 28, 40, 41, 44, 47, 50, 52, 55, 58, 70, 71, 74, 7 ...

能把这串数(取倒数)加起来吗？答案好像是 pi ?

王守恩

northwolves 发表于 2022-12-30 23:01
[1, 2, 4, 5, 7, 8, 10, 11, 14, 17, 20, 22, 25, 28, 40, 41, 44, 47, 50, 52, 55, 58, 70, 71, 74, 7 ...

s=(1 + 1/2 + 1/4 + 1/5 + 1/7 + 1/8)
+(1/10 + 1/11 + 1/14 + 1/17 + 1/20 + 1/22 + 1/25 + 1/28 + 1/40 + 1/41 + 1/44 + 1/47 + 1/50 + 1/52 + 1/55 + 1/58 + 1/70 + 1/71 + 1/74 + 1/77 + 1/80 + 1/82 + 1/85 + 1/88)
+(1/100 + 1/101 + 1/104 + 1/107 + 1/110 + 1/140 + 1/170 + 1/200 + 1/202 + 1/205 + 1/208 + 1/220 + 1/250 + 1/280 + 1/400 + 1/401 + 1/404 + 1/407 + 1/410 + 1/440 + 1/470
+ 1/500 + 1/502 + 1/505 + 1/508 + 1/520 + 1/550 + 1/580 + 1/700 + 1/701 + 1/704 + 1/707 + 1/710 + 1/740 + 1/770 + 1/800 + 1/802 + 1/805 + 1/808 + 1/820 + 1/850 + 1/880)
+(1/1000 + 1/1001 + 1/1004 + 1/1007 + 1/1010 + 1/1040 + 1/1070 + 1/1100 + 1/1400 + 1/1700 + 1/2000 + 1/2002 + 1/2005 + 1/2008 + 1/2020 + 1/2050 + 1/2080 + 1/2200 + 1/2500 + 1/2800
+ 1/4000 + 1/4001 + 1/4004 + 1/4007 + 1/4010 + 1/4040 + 1/4070 + 1/4100 + 1/4400 + 1/4700 + 1/5000 + 1/5002 + 1/5005 + 1/5008 + 1/5020 + 1/5050 + 1/5080 + 1/5200 + 1/5500 + 1/5800
+ 1/7000 + 1/7001 + 1/7004 + 1/7007 + 1/7010 + 1/7040 + 1/7070 + 1/7100 + 1/7400 + 1/7700 + 1/8000 + 1/8002 + 1/8005 + 1/8008 + 1/8020 + 1/8050 + 1/8080 + 1/8200 + 1/8500 + 1/8800)
+(1/10000 + 1/10001 + 1/10004 + 1/10007 + 1/10010 + 1/10040 + 1/10070 + 1/10100 + 1/10400 + 1/10700 + 1/11000 + 1/14000 + 1/17000
+ 1/20000 + 1/20002 + 1/20005 + 1/20008 + 1/20020 + 1/20050 + 1/20080 + 1/20200 + 1/20500 + 1/20800 + 1/22000 + 1/25000 + 1/28000
+ 1/40000 + 1/40001 + 1/40004 + 1/40007 + 1/40010 + 1/40040 + 1/40070 + 1/40100 + 1/40400 + 1/40700 + 1/41000 + 1/44000 + 1/47000
+ 1/50000 + 1/50002 + 1/50005 + 1/50008 + 1/50020 + 1/50050 + 1/50080 + 1/50200 + 1/50500 + 1/50800 + 1/52000 + 1/55000 + 1/58000
+ 1/70000 + 1/70001 + 1/70004 + 1/70007 + 1/70010 + 1/70040 + 1/70070 + 1/70100 + 1/70400 + 1/70700 + 1/71000 + 1/74000 + 1/77000
+ 1/80000 + 1/80002 + 1/80005 + 1/80008 + 1/80020 + 1/80050 + 1/80080 + 1/80200 + 1/80500 + 1/80800 + 1/82000 + 1/85000 + 1/88000)+......
<(1 + 1/2 + 1/4 + 1/5 + 1/7 + 1/8)*1/1 +(1 + 1/2 + 1/4 + 1/5 + 1/7 + 1/8)*4/10 +(1 + 1/2 + 1/4 + 1/5 + 1/7 + 1/8)*7/100 +(1 + 1/2 + 1/4 + 1/5 + 1/7 + 1/8)*10/1000 +......
=(1 + 1/2 + 1/4 + 1/5 + 1/7 + 1/8)*(1/1 + 4/10 + 7/100 + 10/1000 + 13/10000 + 16/100000 +......)
=621/280*40/27=23/7

即:s<23/7=22/7=pi

northwolves

$s=3.1459691497419673>\pi$