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 Next Numbers Please #4 (Posted on 2004-11-16)
What are the next two numbers in this sequence:

1, 2, 4, 16, 26, 42, 57, 512, 730, 1010, 1343, 1872, 2367, 2954

 See The Solution Submitted by Brian Smith Rating: 3.0000 (6 votes)

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 different bases? | Comment 7 of 13 |
(In reply to re: My opinions by GOM)

No matter which base you use (assuming whole numbered bases) the first number is 1.

The second number can not be binary, but it could be base 3.  No matter what, it still refers to the value 2.

The third number, for the same reasons as for the first two numbers, must represent the value 4.

The fourth number becomes tricky:  It can't be in base 6, but it could be in base 7+.  The number 16 in the following bases represent the decimal values in this manner:
base 7  -  13
base 8  -  14
base 9  -  15
base 10 - 16
base 11 - 17
base 12 - 18
base 13 - 19
base 14 - 20

The fifth number can be valid in any base larger than 7 and represent the following decimal values:
base 7  -  20
base 8  -  22
base 9  -  24
base 10 - 26
base 11 - 28
base 12 - 30
base 13 - 32
base 14 - 34

The sixth number could conceivably be back down as a base 5 number (although I doubt it).  Its decimal equivalents are as follows:
base 5  -  22
base 6  -  26
base 7  -  30
base 8  -  34
base 9  -  38
base 10 - 42
base 11 - 46
base 12 - 50
base 13 - 54
base 14 - 58

The last 2 digit number must be in base 8 or larger.  Its decimal equivalents are shown here:
base 8  -  47
base 9  -  52
base 10 - 57
base 11 - 62
base 12 - 67
base 13 - 72
base 14 - 77

One other thing I noticed is that if all the number are in the same base, the first 7 numbers are roughly within a magnitude of difference, and the last 7 numbers are within a magnitude of difference (in respect to number on either side), but from value number 7 to value number 8 there is almost a 10X jump.

Maybe the series of numbers appears in groups of seven?

 Posted by Erik O. on 2004-11-19 17:41:23

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