Let be $P_k = \{n | n = p_1 \cdot \ldots \cdot p_k, p_i = \text{prime numbers not necessary different} |\}$  $P_1 = \text{set of primes }$, $P_2 = \text{integers with exactly two primes factors}$, ....  We know that $|\{n \in P_1, n \leq x \}| = \pi(x) \sim x/\log x$-.   Is there any approximation for $$ | \{ n \leq x | n \in P_k \} | = |\{ n \leq x | n = p_1 \cdot \ldots \cdot p_k, p_i = \text{prime numbers not necessary different}\}|$$
This is my first question.   Let $a_1, a_2,\ldots, a_k$ be natural numbers $\leq n$ with $m$ prime factors.  Let $p_1, p_2, \ldots, p_r$ be the prime numbers $\leq n$.   Let $$C_{m,n} = \frac{\sum_{i=1}^{k}a_i}{\sum_{i=1}^{r}p_i}$$ Does the following limit exist when $m\geq 2$? Obviously, when $m=1$,  $C_{1,n}=1$.  $$\lim_{n\to \infty} C_{m,n}$$  I have some guesses but not defendable enough.  My best,  The FM.
Is the phrase "it's just a matter of semantics" meaningless? I hear this phrase from time to time, and I really don't know what it means.  Two people are debating, and one says "the difference between your position and mine is just a matter of semantics."   This would seem to me to be quite an important difference.  That is, if one person means one thing, and the other person means something else, then they really do have different positions.  If they said "it is just a matter of terminology" or "just a matter of phrasing" I would get it.  But if someone says "we differ only what we mean" ... it kinda makes me chuckle because it seems to be saying that their meaning is not important, as if the argument was primary and the positions they take a secondary consideration.  So I started looking around for examples of this. I guess it seems that there is a popular notion that semantics is about very fine distinctions, such as:   Another interpretation seems to be that semantics is the difference between two ways of saying the same thing:   Maybe it is just a way of stopping an annoying conversation:   And then there is some usage that I simply don't understand:    And also:   Still, to me it is just a meaningless phrase.  Am I missing an important meaning here? Validation Rule on picklist under opportunities There are 2 requirements:    standard users can’t update the status of a record beyond “open won” (so that only SF Admins or auto processes update “registered” and the stages beyond registration. Validation rule for this:    ~   AND(       $Profile.Name &lt;&gt; "System Administrator",      ISCHANGED(StageName),      OR(TEXT(StageName) = "Registered",     TEXT(StageName) = "Pre-Certified",     TEXT(StageName) = "Documentation, Submission",     TEXT(StageName) = "Performance verified",     TEXT(StageName) = "Appeals",     TEXT(StageName) = "Certification/ Closed Won")      )   Requirement 2: Standard users cannot update the status of a record that is already Registered (or post-registered phase). For us, some Opp was already “Registered” but some standard user  was able to change the record to “Open Won.”    I am looking for the validation rule of requirement 2 and which should align with first requirement also.  Thanks, Please guide. What is the 'Badlock Bug'? A user on the Ask Ubuntu General Room posted a link to . After some googling around, all I can find is that it is a mysterious security bug, that uses the same website template as Heartbleed.  I manage Linux Servers, a mysterious security bug does not sit well with me. What exactly is it, and how can I protect my servers from it? How to disable SMS notification during phone call? I call somebody, or somebody calls me, we are talking. Now, during the call I got SMS and I cannot hear anything because the SMS notification is playing.  So how to disable this notification when call is active? Thank you in advance.  Samsung Galaxy Ace 2, Android 2.3.  Note: I am interested in setting this once and for good. Not something I have to remember to switch off each time I make/get a call. Slipstream Windows 7 + Service Pack 1 Why aren't search engines indexing my content? Suppose we have a set  $S=\{a,b\}.$ Obviously, the total number of binary operations on $S$ is the number of all the pairs such that $$*(a,b)=a \, \text{or}  \, b,$$ which gives  $ 2 \cdot 2 \cdot 2 \cdot 2=2^4$ possible ones. Is there a way to find which of these $16$ binary operations are associative without writing all the operation matrices and checking it by hand? What is the highest obtainable resolution of FREE elevation data in the UK? Please add documentation about the redaction tool to the moderator help S tabular with alignment There is a remote Windows server on a private network which I can connect to via Remote Desktop Connection.  I would like to be able to make TCP/IP connections from my computer to other computers on that server's network.  Remote Desktop Connection makes it possible to share printers, drives, and other local resources through the connection.  Is there any way to "tunnel" a TCP/IP connection via RDC?  I'd like something similar to the port-forwarding provided by SSH.  I don't see any way to do this via RDC, but I'm hoping the capability is there and I just don't know about it. Group Membership Test for Permutations Inheriting from a UserControl in WPF I am fairly new to WPF and I am having a problem with inheriting from a user control.  I created a User Control and now I need to inherit from that control and add some more functionality.  Has anyone does this sort of thing before?  Any help would be greatly appreciated.  Thank you Calculate last day of month If you provide 0 as the dayValue in Date.setFullYear you get the last day of the previous month:  d = new Date(); d.setFullYear(2008, 11, 0); //  Sun Nov 30 2008   There is reference to this behaviour at . Is this a reliable cross-browser feature or should I look at alternative methods? When I open the game, the screen stays at the Mojang sign and then crashes. It has something to do with my texture pack, but I can't start the game to change it and it won't let me delete the texture pack folder alone.  I've already deleted the bin and resources folders in the minecraft folder and downloaded the game again. I opened it but the same problem was still there.  How do I properly perform a clean uninstall of the game? I am used to see timestamp everywhere by it's color. The recent timestamps are a little bit reddish everywhere. E.g. in Meta Stack Overflow:  Last edit:        Last seen:        But in Meta Stack Exchange they have same color everywhere even for recent time:  Last edit:        Last seen:         Please make it something like this: I have a 13" MBP late 2009, 2.53 GHz, 4 GB RAM, and on OS X 10.8.3. As soon as I boot up the fans start going at the highest rpm and like I mentioned the kernel_task is always at around 90% when no apps are being run. I read somewhere that I should try safe boot and see the effect but there is none. Any ideas as to what is causing it?  Thanks Prove that the interval $(0, 1)$ and the Cartesian product $(0, 1) \times (0, 1)$ have the same cardinality Prove that the interval $(0, 1)$ and the Cartesian product $(0, 1) \times (0, 1)$ have the same cardinality  using the SB theorem? Also how does one find a bijection on $f:(0, \infty) \to (0,1)$ such that they have the same cardiniality? "Firstly" or "first"? Explanation for "emails"?