[math-fun] Wanted: an _inefficient_ number representation

[Richard Guy]

0, 1, 2, 10, 11, 12, 100, 101, 102, 110, 1000, 1001, ...

is a sqrt(n) example, where the number

abc...k  is to be read as

aT_k + bT_(k-1) + cT_(k-2) + ... + kT_1

where  T_j = j(j+1)/2.

Another example would be

0, 1, 2, 3, 10, 11, 12, 13, 20, 100, 101, 102, ...

which uses squares instead of triangles.

For a cube root example, use cubes:

0,1,2,3,4,5,6,7,10,11,12,13,14,15,16,17,20,21,
22,23,24,25,26,27,28,29,2a,100,101,...    R.

On Mon, 26 Jun 2006, Henry Baker wrote:

> A "unary" representation of a natural number n is
>
> 1 "1"
> 2 "11"
> 3 "111"
>
> etc.
>
> A "positional" representation of a natural number to base b is
>
> d_0+d_1*b+d_2*b^2+...+d_n*b^n
>
> where 0<=d_i<b.
>
> The length of a unary representation of n is O(n), while the
> length of a positional representation of n is O(log(n)).
>
> Are there representations of n that grow as O(sqrt(n)) or O(cbrt(n)) ?
>
>
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