The tag has no wiki summary.

learn more… | top users | synonyms

0
votes
1answer
25 views

Number Theoretic Sum of three variables. Having trouble isolating two of them.

I encountered the following sum: $$\frac{c_{mn-p}}{c_0} = \sum_{d|p}\ \frac{a_{n-d}b_{m-\frac{p}{d}}}{a_0b_0} $$ Where $$a_i=0 \text{ when } i<0 \text{ and } b_j=0\text{ when } j<0\text{ and ...
0
votes
0answers
20 views

Function for the number of divisor of a number [duplicate]

Is there a formula/function that given any $n$ produces the number of divisors of $n$ ? And has that something to do with Euler function?
8
votes
2answers
123 views

Show $\forall\varepsilon>0\,\lim_{n\to\infty}\frac{\#\{\text{positive divisors of n}\}}{n^\varepsilon}=0$

Show that $\forall\varepsilon>0,$ $$\lim_{n\to\infty}\frac{\#\{\text{positive divisors of n}\}}{n^\varepsilon}=0$$ I'm trying to solve this problem for a long time, but I'm really stuck I have ...
14
votes
1answer
361 views

Numbers having in decimal representation no common digits with all their proper divisors

Let us call a positive integer having in decimal representation no common digits with all its proper divisors "a good number". $54$ is a good number : $1,2,3,6,18,27$ $48$ is not a good number : ...
4
votes
1answer
123 views

Euler's totient and divisors count function relationship when $[(\frac{\varphi(n)}{2}+1)\cdot(\frac{\tau(n)}{2}+1)] = n$

I am studying the Euler's totient function $\varphi(n)$ and the divisors count function, $\tau(n)$, also named $d(n)$, and recently opened a question (link here) about the following condition: ...
4
votes
1answer
44 views

Number of Divisors of most numbers

In the book A Comprehensive Course in Number Theory by Alan Baker. The author mentions that even though the average order of $\tau(n)$ is $\log n$, almost all numbers have about $(\log n)^{\log 2}$ ...
0
votes
1answer
26 views

Analogue to superior highly composite numbers for the unitary divisor function

For which positive integers $n$ does there exist a positive real number $\epsilon$ such that $\dfrac{2^{\omega(n)}}{n^\epsilon}\geq\dfrac{2^{\omega(k)}}{k^\epsilon}$ for all $k<n$, and ...
2
votes
1answer
29 views

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 ...
1
vote
1answer
55 views

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.
0
votes
1answer
53 views

$d(n)$ 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 ...
2
votes
2answers
68 views

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 ...
2
votes
1answer
108 views

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 ...
6
votes
1answer
128 views

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 ...
4
votes
4answers
173 views

How many number between 1 and 1000?

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 ...
2
votes
1answer
49 views

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, ...
4
votes
1answer
38 views

Good (asymptotic) upper bound for recurrence over divisors

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 ...
1
vote
1answer
33 views

Prove the divisors pairs

If we arrange the positive distinct divisors of a number A by increasing order, then we get something like: $$1<a_1<a_2<a_3<...<a_{n-2}<a_{n-1}<a_n<A$$How can we prove that ...
0
votes
2answers
55 views

How do I count all values that satisfy X mod N=1 in the range [A,B]

I want to count how many values of x in range [A,B] give remainder of 1 when divided by N. Is there any formula I can apply?
2
votes
2answers
79 views

Is there a way to show that $d(n)$, which counts the number of divisors of $n$ is non-increasing? [closed]

Is there a way to show that $d(n)$, which counts the number of divisors of $n$ is non-increasing? I'm trying to use the Cauchy condensation test to show that $\sum_{n\ge{2}}\frac{d(n)}{n\log^2n}$ is ...
2
votes
4answers
73 views

How to show that $\prod_{d/n} d = n^{\frac{\tau(n)}{2}}$ [duplicate]

set $ n, n \in \mathbb{N}$ and prove that $\prod_{d/n} d = n^{\frac{\tau(n)}{2}}$ ¨I have tried this¨ If $n > 1$ then $n = p_{1}^{\alpha_{1}}\cdot p_{2}^{\alpha_{2}}\cdots p_{k}^{\alpha_{k}}$ ...
5
votes
2answers
114 views

Prove that $\sum \limits_{d|n}(n/d)\sigma(d) = \sum \limits_{d|n}d\tau(d)$

How can I prove: $$\sum \limits_{d|n}(n/d)\sigma(d) = \sum \limits_{d|n}d\tau(d)?$$ Few observations : Left side is a sum function and the right side is a number of divisors function. Both the sides ...
7
votes
3answers
100 views

For any positive integer $n$, show that $\sum_{d|n}\sigma(d) = \sum_{d|n}(n/d)\tau(d)$

For any positive integer $n$, show that $\sum_{d|n}\sigma(d) = \sum_{d|n}(n/d)\tau(d)$ My try : Left hand side : $\begin{align} \sum_{d|p^k}\sigma (d) &= \sigma(p^0) + \sigma(p^1) + ...
6
votes
2answers
2k views

A number is a perfect square if and only if it has odd number of positive divisors

I believe I have the solution to this problem but post it anyway to get feedback and alternate solutions/angles for it. For all $n \in \mathrm {Z_+}$ prove $n$ is a perfect square if and only if $n$ ...
2
votes
1answer
52 views

The number of divisors of any positive number $n$ is $\le 2\sqrt{n}$

How to prove that the number of divisors of any positive number $n$ is $\le 2\sqrt{n}$? I have started something like below: $$ n^{\tau(n)/2} = \prod_{d|n} d$$ But not getting ideas on how to ...
6
votes
1answer
54 views

Sequences where each number is a divisor of one less than the next

Let $N,k$ be fixed. Call a sequence of positive integers $a_1,a_2,\dots,a_k$ good if for each $i$, $a_i$ is a divisor of $a_{i-1}-1$. Consider the set $$S = \{a_i : a_1,\dots,a_k \text{ is a good ...
1
vote
0answers
66 views

Summation of no. of divisors

Let $d(n)$ = no. of divisors of $n$ and $(d(n))^2$ = square of no. of divisors of n. Let $$S(N) = \sum_{n=1}^{N} d(n)$$ and $$S_2(N) = \sum_{n=1}^{N}(d(n))^2$$ $$S(N) = ...
5
votes
2answers
132 views

Finding the summation of the floor of the series identity

I would appreciate if somebody could help me with the following problem: Q: How to proof ? The number of positive divisors of $n$ is denoted by $d(n)$ ...
1
vote
2answers
150 views

Trailing zeroes in factorials: are there any excluded values divisible by 5 other than $5$ and $30$?

I've discovered that when this algorithm for counting zeroes on the end of $n!$ is applied to some $n\in\Bbb{N}$: $$f(n)=\sum_{k=1}^{k:n/5^k\le1}\left\lfloor\frac{n}{5^k}\right\rfloor\notin\{5,30\}$$ ...
1
vote
2answers
70 views

Explain the origin of the number of divisors and sum of divisors formulas.

I know the basic formulas which are: For a number $n = p_1^{a_1} p_2^{a_2} \cdot \ldots \cdot p_k^{a_k}$, we have $d(n) = ( a_1 + 1 )( a_2 + 1 ) \cdot \ldots \cdot (a_k+1)$ and $S(n) = ...
1
vote
1answer
95 views

Bounds on the average of the divisors of natural numbers.

I managed to find an interesting inequality containing the sum of the divisors of a number and the number of them, using AM-GM between each 2 of the divisors of it: $\sigma(n)\geq\sqrt n.d(n)+(\sqrt ...
2
votes
1answer
47 views

Finding $m∈\mathbb N$ such that $d^n(m)$ is not a perfect square for any $n\geq1$

Let , for $k ∈\mathbb N$ , $d(k)$ denote the number of positive divisors of $k$ ; define $d^n (k)$ recursively as $d^1(k)=d(k)$ , for $n\geq1$ , $d^{n+1}(k)=d(d^n(k))$ , how do we find those ...
1
vote
2answers
171 views

Prime factorization, distinct primes

Let $n=p^eq^f$ where $p$ and $q$ are distinct primes and $e$ and $f$ are positive integers. Show that $n$ has $(e + 1)(f + 1)$ distinct factors in $N$, and that the sum of all these factors is ...
0
votes
1answer
57 views

Product of all divisors [duplicate]

Prove that $\prod_{d \mid n}d=n^{v(n)/2}$ where $v(n)$ is the sum of divisors function. We have if $n=p_{1}^{a_{1}}p_{2}^{a_{2}} \dots p_{k}^{a_{k}}$ then $v(n)=(a_{1} +1)(a_{2}+1) \dots (a_{k} ...
8
votes
4answers
469 views

How to prove $ \prod_{d|n} d= n^{\frac{\tau (n)}{2}}$

how to prove: $$ \prod_{d|n} d= n^{\frac{\tau (n)}{2}}$$ $\prod_{d|n} d$ is product of all of distinct positive divisor of $n$, $\tau (n)$ is number (count)of all of positive divisor of $n$