If $f:\mathbb{N}\to\mathbb{N}$ is any strictly increasing function with $f(0)=1$, define the base $f$ notation for natural numbers inductively as follows:

  1. $0$ is represented as $()$ (the empty sequence).

  2. If $n>0$, the representation of $n$ is defined as follows. Let $k$ be maximal such that $f(k)\leq n$. The representation of $n$ is $(f(k))$ (the sequence of length $1$ whose lone entry is $f(k)$) concatenated with the representation of $n-f(k)$.

For example, if $f(n)=10^n$, we get (a version of) decimal notation by the above construction. E.g., the representation of $321$ in that case is $(100,100,100,10,10,1)$.

Now, I'm quite certain this concept is not original (if it were, I'd be shocked to death). Does anybody know who has written about it? (Modulo variations in the exact details, of course)

(See also: "base Fibonacci", which is somewhat similar to the above idea with $f(n)=Fib(n+1)$, but not exactly, as it uses properties of the Fibonacci sequence to make the representations nicer, in a way which does not generalize)

  • $\begingroup$ You could also look at the "type B" expansions in Fritz Schweiger, Ergodic theory of fibred systems and metric number theory (1995). $\endgroup$ – Greg Martin Jan 29 '14 at 22:56

Essentially the system you describe is given at the beginning of section 2 of the paper: "Systems of Numeration" by Aviezri S. Fraenkel in The American Mathematical Monthly, Vol. 92, No. 2 (Feb., 1985), pp. 105-114.

See: http://www.jstor.org/pss/2322638

Other systems are described as well.

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