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Mysterious Sequence (Posted on 2013-09-11) Difficulty: 2 of 5
What is the rule of forming the following sequence : 0,1,10,2,100,11,1000,3,20,101,10000,12,100000,1001,110,4,1000000,21,......

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 5.0000 (1 votes)

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Solution taking the hints | Comment 2 of 3 |
(In reply to Very strong hints by Jer)

The position values from right to left are not powers of some base, but rather successive prime numbers starting with the first prime, 2. And the digits placed in those positions are not the multiple of the place value used in adding to the total value of the number, but rather the power to which the place value is raised in multiplying out to the total value of the number.

Thus as Jer has pointed out, 20 is represented by 102, as 20 = 5^1 * 3^0 * 2^2. In the originals given in the puzzle within the sequence, 9 is represented by 20, as 9 = 3^2 * 2^0.

The primes themselves all begin with a 1 and the rest of the digits are zero, and each has the number of digits as its ordinality in the list of primes.

An encoding algorithm, invoked for the first 45 natural numbers:

 10   for N=1 to 45
 20     Nc=N:Pno=1:S=""
 25     if Nc=1 then S="0"
 30     while Nc>1
 40         Ct=0
 50         while Nc @ prm(Pno)=0
 60             inc Ct
 70             Nc=Nc//prm(Pno)
 80         wend
 90         S=mid("0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ",Ct+1,1)+S
 95         inc Pno
100     wend
110     print N,S
120   next

1      0
2      1
3      10
4      2
5      100
6      11
7      1000
8      3
9      20
10     101
11     10000
12     12
13     100000
14     1001
15     110
16     4
17     1000000
18     21
19     10000000
20     102
21     1010
22     10001
23     100000000
24     13
25     200
26     100001
27     30
28     1002
29     1000000000
30     111
31     10000000000
32     5
33     10010
34     1000001
35     1100
36     22
37     100000000000
38     10000001
39     100010
40     103
41     1000000000000
42     1011
43     10000000000000
44     10002
45     120

  Posted by Charlie on 2013-09-12 00:15:39
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