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Let $m$ be a positive integer and let $f_m(r)=2^{-r}\sum_{i=0}^r\binom{m}{i}$. Clearly $f_m(0)=f_m(m)=1$ and $f_{2r+1}(r)=2^{2r}$. Conjecture: If $m>12$, then the maximum value of $f_m(r)$ for $r\in\{ …

Question: Does anyone know a reference to the following lemmas involving two partitions? (The proofs are not hard, and may well be previously recorded, but where?) First some notation. Let $r$ be a po …

Your first problem has a simple solution. Suppose $p$ is a prime and $(n!)_p$ is the $p$-part of $n!$. Dirichlet proved $(n!)_p=p^k$ where $k = (n-s_p(n))/(p-1)$ and $s_p(n)$ is the sum of the base-$p …

Let $\sigma_p(m,n)$ (resp. $\pi_p(m,n)$) denote the sum of the carries when adding (resp. multiplying) the numbers $m=\sum_{k\ge0}m_kp^k$ and $n=\sum_{k\ge0}n_kp^k$ using base-$p$ arithmetic where $m_ …

I was asked (offline) for a proof of the formula for the sum of the carries $\pi_p(a,b)$ when multiplying $a$ and $b$ in base $p$. Proof. Multiplying the base-$p$ expansions $a=\sum_{k\ge0}a_kp^k$ an …

There are (at least) two explicit formulas. First, $D(n)$ can be shown to be a multiplicative function (this means $\gcd(m,n)=1$ implies $D(mn)=D(m)D(n)$), and the values of $D(p^k)$ for all primes $p …

Does anyone know a reference to the following results, which I can prove, but I suspect may be known. Let $R(n)$ denote the number of primitive $n$th roots of unity with positive real part, and $L(n)$ …

First let me paraphrase the question. Given integers $m,n,k$ each at least 2, set $d:=\max(m,n,k)+2$. Do there exist elements $a,b$ in the symmetric group $S_d$ such that $|a|=m$, $|b|=n$ and $|ab|=k$ …