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Results tagged with nt.number-theory
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user 121059
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
7
votes
4
answers
1k
views
Maximum conjugacy class size in $S_n$ with fixed number of cycles
Context: It is well known that given a permutation in $S_n$ with $a_i$ $i$-cycles (when written as a product of disjoint cycles), the size of the conjugacy class is given by
$$ \frac{n!}{\prod_{j=1}^ …
- 211
3
votes
0
answers
231
views
Is there a closed form for the discrete convolution of $\sigma_1$ and $\sigma_2$?
I am trying to find a closed form for the following sum:
$$\sum_{k = 1}^{n-1}\sigma_1(k) \sigma_2(n-k)$$
where $\sigma_i$ is the sum of divisors and $\sigma_2$ is the sum of squares of divisors.
Le …
- 211
11
votes
3
answers
700
views
Evaluating the sum of $kl^2$ over $p,q,k,l$ such that $pk +ql = n$
I have come across the following sum:
$$\sum_{\substack{p, q, k, l \in \mathbb{N} \\ k > l \\ pk + ql = n}}kl^2$$
and I am trying to simplify it, hoping to get a nice formula in terms of $n$ and som …
- 211