# Questions tagged [decimal-expansion]

For questions about decimal expansion, both practical and theoretical.

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### Representing number, where digits are cubes

Let $n$ is integer and we want to express it in terms of $10$ as $$n=R^3_k10^k+R^3_{k-1}10^{k-1} \cdots+ R^3_0$$ where $R_i\in \{\pm0,\pm 1,\pm 2,\ldots,\pm9\}$ Example : $37= 1^3\times10+3^3$ ...
1 vote
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### Approximate summation formula of time to count numbers from 1 to N

When calculating how much time it takes to count from $1$ to $n$, it is normally used the approximation that it takes about $1s$ to say a number out loud, so it would take $n$ seconds, but there's a ...
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### Proving that there are infinite primes with digit sum 8 in base 10

I recently wrote about a problem I cam up with while thinking about number theory, which you can find on this post. Long story short, I'm trying to prove there are infinite natural numbers such that ...
1 vote
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### Call $n\in\Bbb N$ "balanced" if the sum of its digits equals the count of its divisors. How many "balanced" numbers are there up to $m$?

I recently stumbled across a problem about numbers' divisor count (more specifically, how many positive integers are equal to the square of their divisor count - answer was 2: they are 1 and 9). But I ...
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### Proof of $\inf\left \{ \frac{\mathrm{d} (n^2)}{\mathrm{d} (n)} \; \bigg| \; n \in \mathbb{N} \right \}=0$, where $d(n)$ is the sum of digits of $n$

So I wanted to find the infimum of the set described in the title, and I'm pretty confident on what the subsequence should be to ensure a 0 infimum. But a proof of this eludes me. I tried some funky ...
170 views

### Question about the collection of the prime factors of a fibonacci number

A positive integer $n$ is called pandigital , if every digit from $0$ to $9$ occurs in the decimal expansion of $n$. Conjecture : The largest non-pandigital fibonacci-number (a fibonacci-number with ...