Xnary (like binary but different) counting

Question

I\'m making a function that converts a number into a string with predefined characters. Original, I know. I started it, because it seemed fun at the time. To do on my own. W

Answer 1

Also known as Excel column numbering. It's easier if we shift by one, A = 0, ..., Z = 25, AA = 26, ..., at least for the calculations. For your scheme, all that's needed then is a subtraction of 1 before converting to Xnary resp. an addition after converting from.

So, with that modification, let's start finding the conversion. First, how many symbols do we need to encode n? Well, there are 26 one-digit numbers, 26^2 two-digit numbers, 26^3 three-digit numbers etc. So the total of numbers using at most d digits is 26^1 + 26^2 + ... + 26^d. That is the start of a geometric series, we know a closed form for the sum, 26*(26^d - 1)/(26-1). So to encode n, we need d digits if

26*(26^(d-1)-1)/25 <= n < 26*(26^d-1)/25   // remember, A = 0 takes one 'digit'

or

26^(d-1) <= (25*n)/26 + 1 < 26^d

That is, we need d(n) = floor(log_26(25*n/26+1)) + 1 digits to encode n >= 0. Now we must subtract the total of numbers needing at most d(n) - 1 digits to find the position of n in the d(n)-digit numbers, let's call it p(n) = n - 26*(26^(d(n)-1)-1)/25. And the encoding of n is then simply a d(n)-digit base-26 encoding of p(n).

The conversion in the other direction is then a base-26 expansion followed by an addition of 26*(26^(d-1) - 1)/25.

So for N = 1000, we encode n = 999, log_26(25*999/26+1) = log_26(961.5769...) = 2.x, we need 3 digits.

p(999) = 999 - 702 = 297
297 = 0*26^2 + 11*26 + 11
999 = ALL

For N = 158760, n = 158759 and log_26(25*158759/26+1) = 3.66..., we need four digits

p(158759) = 158759 - 18278 = 140481
140481 = 7*26^3 + 25*26^2 + 21*26 + 3
158759 = H        Z         V       D