# Tagged Questions

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2k views

### Have I found all the numbers less than 50,000 with exactly 11 divisors?

The math problem I am trying to solve is to find all positive integers that meet these two conditions: have exactly 11 divisors are less than 50,000 My starting point is a number with exactly 11 ...
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### Can I presume that this inequality is a good aproximation for a divisor function?

I've used the Lemma 7.9 from page 73 from Krizek, Luca and Somer, 17 Lectures on Fermat Numbers From Number Theory to Geometry Springer CMS (2001) (you can see this page as a Google Book, type here ...
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### Number of solutions to this nice equation $\varphi(n)+\tau(n^2)=n$

How many natural numbers $n$ satisfy the equation$$\varphi(n)+\tau(n^2)=n$$where $\varphi$ is the Euler's totient function and $\tau$ is the divisor function i.e. number of divisors of an integer. I ...
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### proofs of $n\sum_{d|n}\frac{(-1)^{d+1}}{d}=\sum_{d|n,d \text{ odd}}d$

I'm looking for other proofs of the identity : For all integer $n\in \mathbb{N}^{+}$: $$n\sum_{d|n}\frac{(-1)^{d+1}}{d}=\sum_{d|n,d \text{ odd}}d$$ where in the first sums is taken over all ...
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### Find an $n$ not a power of a prime such that $n$ has 51 positive divisors [closed]

Find an $n$ not a power of a prime such that $n$ has 51 positive divisors. I'm not sure where to even start with this question. Any help would be really appreciated.
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### $d(n)$ is odd iff $n = k^2$ [duplicate]

The function $d(n)$ gives the number of positive divisors of $n$, including $n$ itself. For example, $d(25) = 3$ because $25$ has three divisors: $1$, $5$, and $25$. Prove that $d(n)$ is odd if and ...
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### Product of Divisors of some $n$ proof

The function $d(n)$ gives the number of positive divisors of $n$, including n itself. So for example, $d(25) = 3$, because $25$ has three divisors: $1$, $5$, and $25$. So how do I prove that the ...
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### Can someone help me prove that $\tau(n)$ is odd iff $n$ is a perfect square. [duplicate]

Can someone help me prove that $\tau(n)$ is odd if and only if $n$ is a perfect square. So basically I have to prove that $\tau(n)$ is odd iff $n = k^2$ for some integer $k$. $\tau(n)$ is the ...
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### The number of divisors of a number whose sum of divisors is a perfect square

Let $n$ denote a non-prime whose sum of divisors is a perfect square. I have noticed a few surprising facts on the number of divisors of $n$: It is either prime or semi-prime or $27$ in all cases ...
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### How many number between 1 and 1000 satisfy a certain condition?

How many positive integers less than $1,000$ are multiples of $5$ and are equal to $3$ times an even integer? It is simply asking for multiples of $5$ and $6$ Is there a way to do this without ...
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### express the dirichlet series for the sequence d(n)^2 in terms of riemann zeta.

Prove that $$\sum_{n=1}^\infty d(n)^2n^{-s}=\zeta(s)^4/\zeta(2s)$$ for $\sigma>1$ what i did: I already proved this formally, that is, without considering convergence. I use euler products, ...
I am looking for a good (asymptotic) upper bound on the following recurrence relation ($T(0) = 1)$: $$T(n) = \left[ \sum_{1\leq d<n, d|n} T(d) \right] + 1$$ Note that the recursion is only for ...