What is the next number of the series 1, 2, 3, 4, 5, 11, 13, 22?

Answer by Carl-Fredrik Brodda:

According to the OEIS (Online Encyclopedia of Integer Sequences) there are three "commonly" used integer sequences that match the given sequence.
The first one is the digits of the bijective base-k numeration square matrix, read by the antidiagonal starting on row 1, column 10. Quite a mouthful. Essentially, the bijective numerations are the digits by which you can uniquely (hence bijective) describe all non-negative integers (hence, numeration) with as a string of digits in base-k (hence… well, you get it.). By building a matrix of these digits, [math]A(n,k)[/math], we get the square matrix:
1,          1,  1,  1,  1,  1,  1,  1, …
11,         2,  2,  2,  2,  2,  2,  2, …
111,       11,  3,  3,  3,  3,  3,  3, …
1111,      12, 11,  4,  4,  4, 4,  4, …
11111,     21, 12, 11,  5,  5,  5,  5, …
111111,    22, 13, 12, 11,  6,  6,  6, …
1111111,  111, 21, 13, 12, 11,  7,  7, …
11111111, 112, 22, 14, 13, 12, 11,  8, …
I bolded your sequence in the matrix (note that it is read by the anti-diagonal, as promised). In this case, the next number is in fact 121, as you might be able to figure out by spotting the pattern.
The next sequence is a lot more boring. It is simply the square matrix [math]A[/math] where [math]A(n,k)[/math] (row [math]n[/math], column [math]k[/math] is given by the number [math]n[/math] written in base [math]k[/math]. It's quite a lot of fun to draw this matrix on your own, so I won't spoil it for you. Try it out! To get the sequence, you need to again read by the anti-diagonals, and then you get that the next number is in fact 1001.
The last sequence is again the square matrix [math]A[/math] with a certain condition. If [math]\frac{m}{n} = q + \frac{r}{n}, n,m \geq 1, r<n[/math], then [math]A(m,n) = qr[/math]. However, [math]qr[/math] does not mean [math]q[/math] times [math]r[/math] – instead, it is simply [math]q[/math] followed by [math]r[/math], just like 10 does not mean 1 times 0! Hopefully that's clear. Basically, this matrix just deals with remainders when dealing with fractions. It is not too painful. Draw it out to get a feeling for it! Again, to get the sequence, you need to read this matrix by the anti-diagonals (why is that so prevalent?), but if you do draw out the matrix large enough then you will soon spot your sequence! After that, it's child's play. The next number in the sequence is simply 41.
Did you notice the pattern? 41, 121, 1001. It's pretty trivial to see that these are all very special numbers. Think on it for a while. Then scroll down.
Obviously, the connection lies in the duo-decimal system! As always!
41 in base-12 is 49 in base-10, which is a perfect square.
121 in base-12 is 169 in base-10, which is also a perfect square.
1001 in base-12 is 1729 in base-10, which is the smallest taxi-cab number! Wow!
Strangely, OEIS says that there are no known sequences that contain 41, 121, 1001.

What is the next number of the series 1, 2, 3, 4, 5, 11, 13, 22?


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