A number sequence with a specific property

The following increasing number sequence follows a specific number property:

1 2 3 8 10 11 12 13 20 21 22 30 31 68 ...

Find the property and the next two numbers

Hint:

Use squaring!

Hint 2:

hex2dec

• Well, to start, it seems to be rot13(vagrtref, naq vapernfvat). Aug 25 '20 at 19:16
• I hope someone post the old gag of a polynomial interpolating these values. Aug 25 '20 at 19:23
• The point of my previous comment was this: The question doesn't say it's an increasing sequence of integers. So it's possible that that's part of the answer. For example, suppose we were asked "The sequence 0,1,1,2,3,5,8,... follows a certain property. Find the property and the next two numbers". Then the answer is "The sequence follows the property that each element after the first 2 is sum of the preceding 2. The next two numbers are 13 & 21". But suppose we were asked "The sequence 2,3,5,7,11,13,17,... follows a certain property. Find the property and the next two numbers". [continued] Aug 25 '20 at 20:14
• [continued] Then the answer is "The sequence is the set of primes, arranged in increasing order". You'd need to specify it's increasing -- and, more to the point, the question is a bad one, because the answer, "primes, increasing", has two properties that really have nothing to do with one another. It's like asking a sequence whose answer is "primes, that have an odd number of letters in their name" -- why is that interesting? why would any solver think of that? The answer should be cohesive and sensible. In the 2,3,5,7,11 case, a better question would be [continued] Aug 25 '20 at 20:17
• [continued] "Here's a set of numbers, which I've written the first few of in increasing order. Identify the property defining the set, and identify the next two". Then the answer would be "It's the set of primes". So I'm not sure what's going on in our present question. Is it a good question like my Fibonacci one above, or is it a question with a multipart answer whose parts have nothing to do with one another: (2 or more of) "increasing", "integer", and [something else]? I hope it's the former (but will find out when I find or see the answer). Aug 25 '20 at 20:20

They appear to be:

integers whose squares that, when read as a hexadecimal and converted into decimal, result in another square of an integer. That's a bit clunky, so by way of example, 8 is in this sequence because:

8^2 = 64 => 64(hex) = 100(dec) => 100 = 10^2.

Whereas 9 is not in the sequence because

9^2 = 81 => 81(hex) = 129(dec) => 129 is not the square of an integer.

Following this rule, the next two numbers would be:

80 and 100

because

80^2 = 6400 => 6400(hex) = 25600(dec) => 25600 = 160^2

100^2 = 10000 => 10000(hex) = 65536(dec) => 65536 = 256^2

• Congrats, you got it! May 8 '21 at 18:23