The Sesquinary Number Base

"Sesqui" in Latin means "one and a half" and so the sesquinary number base uses 3/2 as its base. It can also be called fractional base 3/2. The numbers formed using this base comprise OEIS A024629:

A024629

\(n\) written in fractional base 3/2.

The initial members of the sequence are:

0, 1, 2, 20, 21, 22, 210, 211, 212, 2100, 2101, 2102, 2120, 2121, 2122, 21010, 21011, 21012, 21200, 21201, 21202, 21220, 21221, 21222, 210110, 210111, 210112, 212000, 212001, 212002, 212020, 212021, 212022, 212210, 212211, 212212, 2101100, 2101101

Figure 1 shows the results more clearly for the numbers 1 to 15:

Figure 1: source

In general, for bases of the form \(a/b\) where \(a\) and \(b\) are integers and \(a \gt b \), the digits that can used range from \( 0\) to \(a-1\) and so for the sesquinary number base the digits 0, 1 and 2 are used. If \(a \lt b \) then the digits \(0\) to \(b-1\) are used.

Let's use \(22_5\) as our first example. This should be the same as \(5_{10} \) and indeed we see that it is:$$  22_5 = 2 \times \frac{3}{2} + 2 =3+2 = 5$$Let's used \(2101_5\) as a second example. This should be equal to \(10_{10} \):$$ \begin{align}  2101_5 &=  2 \times \bigg ( \frac{3}{2} \bigg )^3+ \bigg ( \frac{3}{2} \bigg )^2+0+1 \\ &= \frac{27}{4}+ \frac{9}{4} +1\\ &=10 \end{align} $$The following SageMath code will generate the first 40 numbers (permalink):

def basepqExpansion(p, q, n):

    L, i = [n], 1

    while L[i-1] >= p:

        x=L[i-1]

        L[i-1]=x.mod(p)

        L.append(q*(x//p))

        i+=1

    L.reverse()

    return Integer(''.join(str(x) for x in L))

L=[basepqExpansion(3, 2, n) for n in [0..40]]

print(L)

 I came across this number base in the context of OEIS A081848:

A081848

Number of numbers whose base-3/2 expansion (see A024629) has \(n\) digits.

The initial members of the sequence are (permalink accurate to 5394):

 3, 3, 3, 6, 9, 12, 18, 27, 42, 63, 93, 141, 210, 315, 474, 711, 1065, 1599, 2397, 3597, 5394, 8091, 12138, 18207, 27309

My diurnal age on the date of this post is 27309 and this is how many numbers in base 3/2 has \(n\) = 25 digits. This YouTube video gives a good overview of the different sorts of bases that are possible including negative, algebraic, imaginary and complex number bases. Figure 2 shows a screenshot of the opening frame.

Figure 2: source

Here is another very interesting link titled How Do You Write One Hundred in Base 3/2? Here is an excerpt:

Here’s a problem that I think is hard but might be doable: how many counting numbers are there that have a sesquinary representation that (if we ignore the 0‘s at the front) reads the same forward and backward? (It’s easy to write down palindromic sequences of 0‘s, 1‘s and 2‘s, but the overwhelming majority of them, like 11, don’t correspond to counting numbers.) The biggest sesquinary palindrome I know is four hundred ninety-four, with sesquinary representation 2120010100212. Are there others? Are there perhaps infinitely many others? 

ADDENDUM

Well, 3/2 turned up the very next day, associated with one of the properties of 27310 that turns out to be a member of OEIS A061418:

A061418

a(\(n\)) = floor(a(\(n\)-1)*3/2) with a(1) = 2.

The OEIS comments are particularly interesting:

Can be stated as the number of animals starting from a single pair if any pair of animals can produce a single offspring (as in the game Minecraft, if the player allows offspring to fully grow before breeding again). 

The sequence members are almost identical to those of OEIS A024629 (differences are shown in bold and all differ by being 1 more except for the first member which is 1 less):

2, 3, 4, 6, 9, 13, 19, 28, 42, 63, 94, 141, 211, 316, 474, 711, 1066, 1599, 2398, 3597, 5395, 8092, 12138, 18207, 27310

Here are the members of OEIS A024629 again for easy reference.

3, 3, 3, 6, 9, 12, 18, 27, 42, 63, 93, 141, 210, 315, 474, 711, 1065, 1599, 2397, 3597, 5394, 8091, 12138, 18207, 27309