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Results tagged with integer-sequences
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user 5263
For questions about sequences of integers. References are often made to the online resource oeis.org.
22
votes
Accepted
Is every sequence that looks like an AP really an AP?
For each $n$, the differences $a_{n+1}-a_n$, $a_{n+2}-a_{n+1}$, and $a_{n+2}-a_n$ can only be divisible by powers of $2$ and primes less than or equal to $c$. Since
$$
\frac{a_{n+2}-a_{n+1}}{a_{n+2}- …
- 9,061
5
votes
On vanishing of $p$-adic logarithms
The $p$-adic logarithm can be computed
$$
\begin{align*}
\log_p(2) &= \frac{1}{p-1}\log_p(2^{p-1})\\
&=\frac{1}{p-1}\sum_{n\geq 1}(-1)^{n-1} \frac{(2^{p-1}-1)^n}{n} \\
&\equiv -pF(p)\mod p^2.
\end{ali …
- 9,061
4
votes
Sequences with integral means
Here are some bounds on the stable value $r(n)$, as well as the number of terms of the sequence that need to be calculated. The short version is that
$$
\frac{\sqrt{2}}{2}\sqrt{n}+O(1)\leq r(n)\leq \f …
- 9,061