All Questions
Tagged with divisor-sum multiplicative-function
23
questions
1
vote
1
answer
83
views
Prove that there are infinitely many natural number such that $σ(n)>100n$
The problem is as follows: Prove that there are infinitely many natural numbers such that $σ(n)>100n$. $σ(n)$ is the sum of all natural divisors of $n$ (e.g. $σ(6)=1+2+3+6=12$).
I have come to the ...
- 19
1
vote
1
answer
144
views
On a conjecture involving multiplicative functions and the integers $1836$ and $137$
We denote the Euler's totient as $\varphi(x)$, the Dedekind psi function as $\psi(x)$ and the sum of divisors function as $\sigma(x)$. Are well-known arithmetic functions, see the corresponding ...
4
votes
1
answer
107
views
On a conjecture involving multiplicative functions and the integers $1836$ and $136$
We denote the Euler's totient as $\varphi(x)$, the Dedekind psi function as $\psi(x)$ and the sum of divisors function as $\sigma(x)$. Are well-known arithmetic functions, see Wikipedia. I would like ...
3
votes
1
answer
107
views
The mean square of $d_k(n)$
Let $d_2(n)=d(n)$ be the divisor function, and let $$d_k(n)=\sum_{d_1\cdots d_k= n}1=\sum_{m\cdot l= n}d_{k-1}(m).$$ Can anyone point me to a reference to the size of the error term when approximating
...
- 1,662
1
vote
0
answers
131
views
Proving a relation between sum of reciprocal of divisors and $\sigma(n)$
Prove that $\sum_{d\mid n}\dfrac{1}{d}=\dfrac{\sigma(n)}{n}\ \forall\ n\geq 1, n\in \mathbb
Z$
My question and my approach is a lot similar to this question and a bit different from this question ...
- 1,901
1
vote
1
answer
93
views
Is $\frac{\sigma(n)}{{2n}}$ submultiplicative or supermultiplicative?
This question is an offshoot of this earlier post.
It is known that the abundancy index
$$I(n) = \frac{\sigma(n)}{n}$$
is a multiplicative function. (Note that the divisor sum $\sigma$ is also ...
- 8,010
2
votes
1
answer
107
views
Is this function multiplicative and if so what is its value at prime powers?
For odd numbers $n$ let:
$$a(n) = \sum_{d^2|n} d \frac{\sigma^*(n/d^2)}{2^{\omega(n/d^2)}}$$
where $\sigma^*(k) = $ sum of unitary ($\gcd(d,k/d)=1$) divisors of $k$ and $\omega$ counts the prime ...
user276611
0
votes
1
answer
278
views
If an arithmetic function $f$ satisfies $f(mn) \leq f(m)f(n)$ (whenever $\gcd(m,n)=1$), is $f$ weakly multiplicative or submultiplicative?
From the preprint On sums of the small divisors of a natural number (Lemma 1, page 2) by Douglas E. Iannucci:
We observe here that the function $a(n)$ is not multiplicative. It is, however, ...
- 8,010
1
vote
1
answer
81
views
Are there (restricted) instances when the deficiency and sum-of-proper-divisors functions are multiplicative?
A function $f : \mathbb{N} \rightarrow \mathbb{Q}$ is said to be multiplicative if
$$f(ab) = f(a)f(b)$$
whenever $\gcd(a,b)=1$.
It is known that the sum-of-divisors function
$$\sigma(x) = \sum_{d \...
- 8,010
1
vote
1
answer
316
views
Multiplicative functions and the sum of all divisors: $\sum_{d\mid2020}{\sigma(d)}$
Doing more practice for my final and I need some help with the following:
Evaluate:
$$\sum_{d\mid2020}{\sigma(d)}$$
where $\sigma(n)$ is the sum of all divisors of n.
The hint given specifically ...
- 251
3
votes
2
answers
813
views
About The Sum of Positive Divisors of $n$
The question says:
Find the smallest positive integer $n$ so that $\sigma(x)=n$ has no solution, exactly two solutions, exactly three solutions.
I could not come up with a good way to solve this ...
- 1,150
2
votes
3
answers
487
views
How to do a sum of a function over all divisors of an integer with Maple?
I'd like to define sumdiv in Maple such that this:
with(numtheory);
f:=x->x^2;
sumdiv(f(d)*mobius(100/d), d=1..100);
would ...
- 1,561
2
votes
1
answer
138
views
Show that $\sigma(n)$ is equally often even and odd
Let $\sigma(n)$ be the divisor sum of $n$:
$$ \sigma(n) = \sum_{d|n} d. $$
I was interested in the parity of $\sigma(n)$ and tried to check whether $\sigma(n)$ is unexpected often even for odd $n$ ...
- 908
5
votes
2
answers
166
views
Can $\sigma(p^k)=2^n$ for some $k>1$?
If $k=1$ then the only solutions to
$$
\sigma(p^k)=2^n
$$
are when $p$ is a Mersenne prime (of course $p$ is restricted to the primes). Is there a solution for larger $k$? It doesn't seem so but I can'...
- 32.3k
2
votes
2
answers
143
views
Closed form for $\sum\limits_{d|n}\sigma(d)$?
Is there any closed form for $$\sum\limits_{d|n}\sigma(d)$$? I knew that $\sum\limits_{d|n}1=\sigma(n)$ therefore we must have $\sum\limits_{d|n}\sigma(d)=\sum\limits_{d|n}\sum\limits_{r|d}1$ but how ...
- 761