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3 questions
11
votes
1
answer
891
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The maximum of the preimage of [1,x] through Euler's totient function
A friend of mine and I have shown the following:
"For each $x \geq 1$ let $m := m(x)$ be the greatest positive integer such that $\varphi(m) \leq x$, where $\varphi$ is the Euler's totient function.
...
user40023
9
votes
0
answers
358
views
Being even or odd in the product expansion $\prod(1+x^k+x^{k+1})$
Consider the generating function of "partitions with distinct parts"
$$\sum_nQ(n)x^n=\prod_k(1+x^k).$$
It's known that
$$\left[\prod_k(1+x^k)\right] \mod 2=\prod_m(1-x^m)=\sum_{j\in\mathbb{Z}...
- 43.2k
3
votes
2
answers
876
views
Asymptotics for the number of partitions of $n$ into odd prime parts
Hello!
I am interested in the asymptotic behavior of the function $p_o(n)$ defined as the number of partitions of $n$ into odd prime parts A099773 - http://oeis.org/A099773 .
I couldn't find any ...
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