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0
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2answers
32 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?
0
votes
0answers
15 views

Number of multiples of 6 between -600 and 3400

Could someone please verify if I did this correctly? The number of multiples of 6 between 3400 and -600 should be the floor of (3400 / 6) - the floor of (-600 - 1 / 6) + 1 (to account for zero. Is ...
2
votes
2answers
66 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
58 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 $n = p_{1}^{\alpha_{1}}\cdot p_{2}^{\alpha_{2}}\cdots p_{k}^{\alpha_{k}}$ ...
3
votes
2answers
357 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
40 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
50 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
57 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) = ...
4
votes
2answers
103 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 $d(n)$ ...
0
votes
2answers
131 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
63 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
86 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
46 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 ...
0
votes
1answer
44 views

Product of all divisors

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} ...
6
votes
4answers
224 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 $n$