# Tagged Questions

3k views

### Does every prime divide some Fibonacci number?

I am tring to show that $\forall a \in \Bbb P\; \exists n\in\Bbb N : a|F_n$, where $F$ is the fibonacci sequence defined as $\{F_n\}:F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2}$ $(n=2,3,...)$. How can ...
172 views
+200

### Divergence for $p$ prime numbers and convergence for $m$ composite numbers

Is there exist a sequence $(a_n)_{n\in \mathbb N} \in \mathbb C^{\mathbb N}$ such that : For all $p$ prime numbers the serie $\displaystyle \sum_{n\in \mathbb{N}} a_n^p$ diverge, and for all ...
52 views

124 views

### Value of sum with primes

Can anyone tell me the value of sum $\sum_p\left(\log p(\frac{1}{2p}-\psi(\frac{p+2}{2})+\psi(\frac{p+1}{2})\right)-\sum_n\frac{(\log(2^n)}{2^n}$ where $p$ ranges over prime powers and $n$ ranges from ...
157 views

### Calculation involving $\int_2^x \frac{dx}{\log x}$

Background (skip to the gray if you prefer). In Legendre's 1798 work on number theory he conjectured that $\pi(x)\sim \frac{x}{\log x - A}$ in which he proposed that $A = 1.08366.$ Gauss disputed the ...
58 views

### Infinite series with prime number [duplicate]

I know the $\sum_{n=1}^\infty \frac{1}{\text{Prime[$n$]}}$ does not converge, but what about the following series? $$\sum_{n=1}^\infty\frac{1}{\text{Prime[Prime[n]]}}$$ (Where $\text{Prime[$n$]}$ ...
77 views

### Sum containing primes

Can anybody compute the value of $$\sum_p\sum_{k=2}^\infty\frac{\log(p^k)}{k}-\sum_p\sum_{k=2}^\infty\frac{\sum\limits_{p^n<k}\log(p^n)}{k(k+1)}$$ I have tried a lot but cannot think about the ...
177 views

### “Interesting” Sequences

Well, here's a question i myself made up and i thought it's interesting if i share it with everyone. We call a sequence of natural numbers (for example $a$) Interesting if (all three must be true): ...
79 views

### Discrepancy between terms of sum and sum

My question is why the following happens, and whether we can correct (2) below to account for an errant factor of 2. By a slight generalization* of the argument of this problem we have I think that ...
87 views

### Series equivalent to $\sum p_k$

Looking at a theorem of Chebyshev, I noticed that $$\sum_{n=0}^{\infty} \sum_{p_k < n} \frac{(\log p_k)^n}{n!} = 2 + 3 + ...+ p_k.$$ Proof. Letting $x = \log p_k$ and writing out the expansion of ...
101 views

### Does $f(n)\sim g(n)$ imply $\lim_{k\to\infty} \frac{1}{k} \sum_n f(n)/g(n) = 1$?

Is it true that $$\lim_{k\to\infty}\frac{1}{k}\sum_{n=1}^k \frac{f(n)}{g(n)} = 1 \leftrightarrow f(n)\sim g(n).$$ My thought: $f(n)\sim g(n) \rightarrow \frac{1}{k}\sum \frac{f(n)}{g(n)} = 1$ since ...
147 views

### Limit of $\sum\frac{1}{p(\pi(n))}$

Let $p(n)$ be the nth prime and $\pi(n)$ the number of primes not exceeding n. I wonder if we can show that $$\tag{1} S = \sum_{n= 2}^k \frac{1}{ p (\pi (n))} \sim \log k.$$ We know by comparison ...
201 views

429 views

33 views

### Is there a pattern (or a name and expression for the pattern) of the intervals between all primes?

With the recent interest in Mersenne primes, I got thinking whether there was any mathematical expression for the pattern of intervals (or sequence composed of interval lengths) between ordinary prime ...
129 views

### Sequence involving primes of form $n^2 + n+1$

Looking at prime numbers $p_i$ of the form $n^2+n+1$ and the derived expression $$1 - \prod_{i=1}^{j}\frac{(p_i-1)}{p_i}$$ it seems (I do not claim it and do not see why it should be true) that ...
Here is a sequence; experimental mathematics : $2, 2, 2, 11, 11, 254908033,...$ we could define as : Least primes such that $((p+1)(nextprime(p)+1))-1$ is prime and ...
I.e., is there a sequence of primes whose decimal expansions have the following form: $$a_1,\ a_1a_2,\ a_1a_2a_3,\ a_1a_2a_3a_4, \dots$$ What about with the order of the digits reversed, so each ...