How to replace addition with multiplication to find the next integer value?

Question

Sorry in advance for my lack of mathematical knowledge, I am very new to it.

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Yesterday, I posed this question to myself:

"In a world without addition or subtraction, how could we derive the next value in the sequence of natural numbers from $1\to\infty$ with a step size of $1$?"

This lead me to the idea of multiplication to find the next value in a sequence. After analyzing the multipliers between each natural value using: $$ \frac{(n+1)}{n} $$

I noticed the pattern of this sequence starts at the high values of $2$ and $1.5$, then converges to a value of $1$.

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My two questions:

• Is it right to assume that the sequence of multipliers should have a more predictable sequence?
• Are there more elegant ways of producing the next natural number without addition or subtraction?

Answer 1

With the function $2^n$ and it's inverse, $\log_2$ available, $n+1=\log_2(2\cdot2^n)$.

Answer 2

If we were allowed to use floor and ceiling functions in this strange world without $+$ and $-$, then perhaps we could use the following function which would generate the next integer after $n$:

$$f(n)=\lceil{(n\times\frac{\lceil n\sqrt2\rceil}{\lfloor n\sqrt2\rfloor})\rceil}$$