# What is the law behind this sequence?

Can you figure out the rule that produces this sequence?

10, 10, 11, 12, 15, 16, 21, 21, 23, 207, 207, 211, 213, 213, 215

What is the next term?

Here's a hint:

Even if it seems otherwise, the sequence does not contain repeated numbers.

• Re the hint, it does. 4 times. Surely the sequence is what you gave, not what it was generated from. Aug 27, 2016 at 15:07
• @bg6471: It contains repeated digit sequences, but not repeated numbers. A digit sequence is not a number. Aug 27, 2016 at 15:35

the prime numbers written base n
for n = 2, 3, 4, ... and you reserve two digits as base n > 10

for example

2 = 10 base 2
3 = 10 base 3
5 = 11 base 4
7 = 12 base 5
11 = 15 base 6
13 = 16 base 7
17 = 21 base 8
19 = 21 base 9
23 = 23 base 10
29 = 207 base 11 (reserving two digits as using base 10 for two digit numbers)
...
and so on until

the next term in the sequence after 215 is

302 as it is 53 written in base 17 using this notation

• You got everything right except the final number. You gave a wrong base, and even if the base were right, the value would not be valid in the base you gave. Aug 27, 2016 at 13:08
• Thanks celtschk for this and yes you are right - I have corrected the final term!
– Tom
Aug 27, 2016 at 13:12
• Now everything is correct. Well done! Aug 27, 2016 at 13:14