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 Mysterious Sequence (Posted on 2013-09-11)
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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 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      02      13      104      25      1006      117      10008      39      2010     10111     1000012     1213     10000014     100115     11016     417     100000018     2119     1000000020     10221     101022     1000123     10000000024     1325     20026     10000127     3028     100229     100000000030     11131     1000000000032     533     1001034     100000135     110036     2237     10000000000038     1000000139     10001040     10341     100000000000042     101143     1000000000000044     1000245     120`

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

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