Is there an equation $f(k,m,n)=n$ where $n$ is the number of applications of the Ackermann function needed? Using the definition of the Ackermann function on , (sidenote, great paper): $$  \alpha(k,m,n) = \begin{cases} m+n,  &amp; k=1 \\ m, &amp; n=1 \\ \alpha(k-1,m,\alpha(k,m,n-1)),&amp; \text{otherwise} \end{cases} $$  Is there some function $f$ such that $f(k,m,n)$ returns the number of times $\alpha$ is used to calculate $\alpha(k,m,n)$?  Some trivial values would be $f(1,m,n)=f(k,m,1)=1$, and $f(2,2,2)=2$, found manually by counting the number of $\alpha$s in the expanded form of $\alpha$.  Note: There is neither a tag for "ackermann-function" or "recursion", I lack the privilege to create these but I feel they would have some limited use.
Ackermann function - how to calculate the number of times it calls itself Purely for my own amusement I've been playing around with the .  The Ackermann function is a non primitive recursive function defined on non-negative integers by:  $A(m,n) = n+1$, if $m=0$  $A(m,n) = A(m-1,1)$, if $m&gt;0$ and $n=0$  $A(m,n) = A(m-1,A(m,n-1))$, if $m&gt;0$ and $n&gt;0$  I'm not interested in values of the function itself, rather I'm interested in the number of recursions the function takes to arrive at a value. For example:  $A(1,2) = A(0,A(1,1)) = A(0,A(0,A(1,0))) = A(0,A(0,A(0,1))) = A(0,A(0,2)) = A(0,3) = 4$  That's 6 steps, so if we define $RA(m,n)$ as the number of recursions to calculate $A(m,n)$, then $RA(1,2) = 6$.  I made a short Excel macro to calculate $RA(m,n)$ for any given $m$ and $n$. The problem I have is that $RA(m,n)$ grows even faster than $A(m,n)$ and I quickly run out of stackspace. The table below shows $RA(m,n)$ for very small values of $m$ and $n$. The numbers shown for $RA(3,10)$ and $RA(4,1)$ are the number of steps before crashing!    A formula for $RA(1,n)$ is trivial, and it is relatively easy to see that $RA(2,n) = 5(n+1) + 4n(n+1)/2$.  So my questions: can anyone see an explicit formula for $RA(3,n)$? More generally, does a formula exist for $RA(m,n)$, in the same way that $A(m,n)$ can be expressed using Knuth arrows?   For the low values shown of $m$ and $n$, $RA(m,n)$ looks about order of magnitude larger than $A(m,n)$. Can we prove that $RA(m,n) &gt; A(m,n)$ for all $m,n$. Intuitively, it should be.  For bonus points, can $RA(m,n)$ be expressed in terms of $A(m,n)$?
I have a php file which I will be using as exclusively as an include. Therefore I would like to throw an error instead of executing it when it's accessed directly by typing in the URL instead of being included.  Basically I need to do a check as follows in the php file:  if ( $REQUEST_URL == $URL_OF_CURRENT_PAGE ) die ("Direct access not premitted");   Is there an easy way to do this? Which badges do not cause a notification for experienced users? Java obfuscators I'm looking for a good Java obfuscator.  I've done initial research into the following Java obfuscators: proguard, yguard, retroguard, dasho, allatori, jshrink, smokescreen, jobfuscate, marvin, jbco, jode, javaguard, jarg, joga, cafebabe, donquixote, mwobfu, bbmug, zelix klassmaster, sandmark, jcloak, thicket, blufuscator, and java code protector.  I tried proguard and it has a really nice GUI, seems really stable, and seems to be the most popular, but it seemed to not like some enumeration on a referenced jar file (not within the code I was trying to obfuscate) which was weird.  Yguard seems to require some interaction with ant, which I didn't know too much about.  What is a good java obfuscator?  It doesn't need to be free, it just needs to work well and be easy to use. Use induction to prove a product of sums of squares is a sum of squares Which Marvel TV episodes are related to the MCU movies? I've kept up to date with the movies from the MCU, but I am grossly out of date with the TV series, of which I think there is only Agent Carter (not as relevant for the movies I think, besides that scene in Captain America 2), and S.H.I.E.L.D. (which from what I've seen seems more relevant, with  coming in phase 3).   I've gained what I can from . But I can't tell which of the episodes listed are relevant to the movies (clearly, it seems that none are essential), and which movies are relevant to the episodes.  So which episodes are related? What's a recommended alternative to Quickbooks pro? I've just installed Ubuntu to replace Windows. I can't seem to find a good alternative to  to run on Ubuntu. Any help would be appreciated. How do I change my username? How to fix broken taskbar icons in Windows 7 Launchers for installed snap applications (including Ubuntu Software) disappeared after installing GNOME Software Is there anything we can do to locate companions in the world easier? Keeping track of where everyone is is hard.  I could be out in the wasteland while Dogmeat will let me know he found something. That's great, but then it's a chore trying to find him a lot of times. Sometimes, he'll even run ahead and attack enemies and I'll lose sight of him. Only if I leave an area sufficiently far enough, he'll come back and meet with me which isn't always possible. I know I can listen out for him but visual indicators would be better. But none of this works if I sent him to a settlement but needed to find him and he hasn't returned yet.  Is there anything we can do to make locating companions in the world easier? I'm writing a report, in LaTeX. And I need to include, as appendices, other latex documents I have written. I am currently doing this using pdfpages and it works. However, the inserted PDF has no page numbers and it overlays my base documents page numbering, resulting in no page numbers on the PDF pages. My questions:   How to solve the above problem. Is there a better way to do this. Show that a Cauchy sequence has a fast-Cauchy subsequence How to get things off your head in Minecraft Mo' Creatures mod I installed the Mo'Creatures mod into Minecraft, then made a fish bowl. After catching a fish I went to put it on top of a bookshelf in my room, but instead of placing it, the bowl just dropped at my feet. When I tried to pick it up my character put it on his head. Now I'm stuck with a fish bowl on my head and I can't find any mention anywhere of how to get it off. There are a number of other animals from the mod that you can put on your head for some effect, but no mention of how to remove them pretty much anywhere. Short story identification: society partitioned off by high wall Is there a word in English to qualify the highest-ranked student in a class in a year?  We are talking here about higher education, especially in the context of curricula that keep a given set of students over the course of several years, being ranked at the end of each year, possibly with competitive admission for the following year. The ranking is a weighted average of all marks obtained over the year.  I am looking for a single word, or expression, to quickly designate the one student with the highest ranking. If there is no such thing in English, then let's try to get the most concise and idiomatic definition.  In all cases, it is not emotionally connoted, i.e. not pejorative nor bragging.  For French speakers: I am looking for the translation of “major de promotion”.    EDIT: additional, not as much of a pure “word request” question: is this anything you should mention on a curriculum vitæ in the English-speaking world? If so, please mention whether the given word is appropriate for such a use&nbsp;&nbsp;:) I'm stuck in a teleporting loop. What can I do? I set a command block to /tp. It was supposed to activate only if you had 0 time left, and to activate the command block with a hopper clock to keep giving it power.   But it is not checking if you have 0 time left, so it keeps teleporting me. I can't get close enough to the command block to break it, so I'm stuck in a neverending teleport loop.  Any ideas? What's wrong with this Wheatstone bridge? I was trying to solve this unbalanced Wheatstone bridge and found that the current $i$ in the mid wire is 9A .This result was achieved by using node voltage method.    Then I rearranged the same circuit to the form shown in the second image.  and at this point I flipped the right half of the circuit so as to obtain the equal resistors on the same side.    now I again rearranged the circuit as shown.    Here, the final circuit  is a balanced Wheatstone bridge and thus the current i must be 0.  So clearly something went wrong in there.  I saw a post  before this one just to make sure that there is no mistake in rearranging the circuit in the way I have done.   It was answered by the user PhillipWood as being correct and was unfortunately put on hold.  So can someone please point out what exactly went wrong with the above method. Make links posted by users open in a new window Most of the time when I follow a user's link, I just want to see their site, look it over, then return to answer their question (not leave SO completely and not come back). Could we make the links users post open in a new window please?  EDIT: I originally wanted this implemented because it's what some other sites did, but I now definitely prefer SO's way. If I want to open a new tab, I middle-click/control-click (or whatever); if not, I just click. I'm in control of what happens. Would never want to go back now. Request root privilege from within a script Suppose the gcd (a,b) = 1 and c divides a + b. Prove that gcd (a,c) = 1 = gcd (b,c) I am lost.   So far...  If $\gcd (a,b) = 1$, by Bezout's Formula  $ax + by = 1$  If $c|(a+b)$, then $cf = a+b$  Then,  $a (x-y) + cfy = 1$   $b(yx) + cfx = 1$  Am I on the right track? Any suggestions?