ETA: In case someone notices the similarity, this question is not a duplicate of .     Which statement is true about the sequence $\{f(n)\}$:    $$ \left\{  \begin{array}{l} f(1)=1 \\  f(2n)=f(n) \\  f(2n+1)=f(n)+f(n+1) \end{array} \right.  $$   a)All prime numbers appear in $\{f(n)\}$   b)All Mersen primes appear in $\{f(n)\}$   c)A finite set of primes appear in $\{f(n)\}$   d)Just primes of the form $4t-1$ appear in $\{f(n)\}$     It's seen that $f(2n)=1$ but I'm not sure that $f(2n+1)$ generates which primes...
An arithmetic function is defined as follows:$f(1)=1$,  $f(2k)=k$ and $f(2k+1)=f(k)+f(k+1)$. When (for which $k$) is $f(k)$ even?  While it is obvious that $f(4n-1)=f(4n)=2n$, therefore $f(k)$ is even for $k=4n$ and $k=4n-1$, which is possible to prove using induction (isn't it?), I don't know how to prove the converse.  I tried to solve the recurrence or make use of constant difference between certain elements of the sequence $\ldots$ What am I missing?  Thank you.
So, I'm working on a control station that has buttons. It is a one-mesh creation, that means, I don't want to create buttons and things like that separated from the main mesh, but as you can see, I have problems. I would like to make the highlighted parts to be one with the main part around them. I've tried everything, and I have this problem on multiple creations. Any idea?  Update: I've added another example of the problem. I want them to be one face, instead of two. (2nd picture) Transit Visa in Hong Kong for 15 hour layover between India and USA? "Images with street signs" CAPTCHA impossible to solve On June 13th 2017 all SharePoint calculated columns with HTML markup stopped working.   Basically the fields now render text (regardless of setting the data type to number). The gist of the notice from Microsoft () is:     In Microsoft SharePoint lists and libraries, calculated fields that   are running in the classic UI mode can be used to display results   based on Excel-like formulas. This is a long standing capability, and   is documented in Calculate data in lists or libraries.      Some users have added HTML markup or script elements to calculated   fields. This is an undocumented use of the feature, and we will begin   blocking execution of custom markup in calculated fields in SharePoint   Online from June 13, 2017 onwards. We are also providing this as a   configurable option for on-premises in SharePoint Server 2016 and   SharePoint Server 2013 via the June 2017 and subsequent Public   Updates.   As we don't have on premise SharePoint (online), we have to find an immediate work around, given that this impacts our processes and thus, business.   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Here's what it says:     Considering a one-dimensional quantum system in a potential $V(x)$ with degenerate states $|\psi_1\rangle$ and $|\psi_2\rangle$ corresponding to the same energy eigenvalue $E$, writing the time-independent Schrödinger equation for the system:       $$-\frac{\hbar^2}{2m}\frac{ \partial^2\psi_1}{ \partial x^2} + V\psi_1 =E\psi_1$$       $$-\frac{\hbar^2}{2m}\frac{ \partial^2\psi_2}{ \partial x^2} + V\psi_2 =E\psi_2$$   Multiplying the first equation by $\psi_2$ and the second by $\psi_1$ and subtracting one from the other, we get:       $$\psi_1\frac{d^2}{d x^2}\psi_2-\psi_2\frac{d^2}{d x^2}\psi_1=0$$   Integrating both sides       $$\psi_1\frac{\partial \psi_2}{\partial x}-\psi_2\frac{\partial \psi_1}{\partial x}=constant$$   In case of well-defined and normalizable wave functions, the above constant vanishes, provided both the wave functions vanish at at least one point, and we find: $\psi_1(x)=c\psi_2(x)$ where $c$ is, in general, a complex constant. So, the two degenerate wave functions are not linearly independent. For bound state eigenfunctions, the wave function approaches zero in the limit $x\to\infty$ or $x\to-\infty$, so that the above constant is zero and we have no degeneracy.   Here are the parts I don't understand:   How does integrating $\psi_1\frac{d^2}{d x^2}\psi_2-\psi_2\frac{d^2}{d x^2}\psi_1=0$ yield $\psi_1\frac{\partial \psi_2}{\partial x}-\psi_2\frac{\partial \psi_1}{\partial x}=constant$.  Without having explicit formulas for $\psi_1(x)$ and $\psi_2(x)$, how do we just apparently pull them both out of the integral? I don't understand how the constant in the last equation vanishes just because $\psi_1$ and $\psi_2$ vanish at infinity. Even if it does vanish, how does that imply $\psi_1=c\psi_2$? How to programmatically click a button in WPF? This started to happen immediately after I had rebooted the first time after doing a system upgrade. It first starts with a dialogue that says "System program problem detected". Then when I try to hit 'report problem' not much happens. I am led through a dialogue that always ends up the problem cannot be solved.  I am aware this is not a lot of information, however I'm not sure which information I need to publish and how should I obtain it to debug this problem.  Here's a screenshot! How do I recover my Google account (or Gmail) password or username? I have forgotten the password or username for my Google account, or I'm unable to access my account for other reasons.  How can I recover my account?  (This Q&amp;A is meant as a general description for all Google and Gmail username/password questions. See: ) Difference between administrative and user mode of MiKTeX How to disable internet search results in start menu post Creators Update? It  to disable internet search results in start menu searches before the Windows 10 Creators Update, but I've just clean installed a new Windows 10 Pro image and that setting is gone. Is there some way to disable it through editing the registry/etc? “situation where” vs. “situation in which” It looks like if I load dynamic content using $.get(), the result is cached in browser.  Adding some random string in QueryString seems to solve this issue (I use new Date().toString()), but this feels like a hack.   Is there any other way to achieve this? Or, if unique string is the only way to achieve this, any suggestions other than new Date()? What would be your preferred way to concatenate strings from a sequence such that between every two consecutive pairs a comma is added. That is, how do you map, for instance, ['a', 'b', 'c'] to 'a,b,c'? (The cases ['s'] and [] should be mapped to 's' and '', respectively.)  I usually end up using something like ''.join(map(lambda x: x+',',l))[:-1], but also feeling somewhat unsatisfied. Generate two random numbers with constraints What is the definition of layer in neural network? Which Harry Potter works are considered canon? Question about how to interpret arbitrary elements I have a question about how to interpret arbitrary elements. When we say:  &quot;let $x$ be an arbitrary integer&quot;  &quot;let $x$ be any integer&quot;  &quot;let $x$ be an integer&quot;   at the beginning of proofs, are we saying:  &quot;let the symbol $x$ represent one of the integers, although it is not clear which specific integer the symbol $x$ represents&quot;.  If not, please let me know how I should think about the above statements. Thank you for your time.  Edit: I had a follow up question to the one I asked earlier. What do mathematicians mean by “arbitrary element” or “any element”. I can understand when mathematicians say “let $x$ be $4$” as this means “the symbol $x$ represents $4$”. Could someone please elaborate on why the specific terms “arbitrary” and “any” are used? With regard to the theory that each time a wave function collapses the universe  so that each possible outcome really exists - where does all the energy required to create all the new universes come from?