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I have a couple of questions about numbers that satisfy a certain property. The numbers have the property that $\overline{a_1a_2a_3a_4\dots a_n}=a_1+a_2^2+a_3^3...a_n^n$ where $\overline{a_1a_2...a_n}$ is the decimal representation of the number. These are numbers like... $$518=5+1^2+8^3\\135=1+3^2+5^3 \\175=1+7^2+5^3\\598=5+9^2+8^3$$

  1. Do these numbers have a name?
  2. Are there infinitely many of them?
  3. Is there any formula to calculate them?
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1 Do these numbers have a name?

I am not aware of these numbers having a special name, so let's call them Joao numbers for now.

2 Are there infinitely many of them?

Note that $a_1+a_2^2+a_3^3+\cdots + a_n^n \le 9+9^2+9^3+\cdots + 9^n = \dfrac{9}{8}(9^n-1)$

and that $\overline{a_1a_2a_3\cdots a_n} = a_110^{n-1} + a_210^{n-2} + \cdots + a_n \ge 10^{n-1}$.

So Joao numbers exist only for $n$ such that $10^{n-1} \le \dfrac{9}{8}(9^n-1)$, i.e. $n \le 22$.

Hence, there is no Joao number larger than $10^{22}$, and thus, there are finitely many Joao numbers.

3 Is there any formula to calculate them?

One way to compute all Joao numbers is to check every number up to $10^{22}$. However, this is severely computationally intensive. I would not recommend this unless you have a supercomputer.

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  • $\begingroup$ I just found something: oeis.org/A032799 is this sequence $\endgroup$ – Joao Sep 21 '14 at 4:06

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