There is an identity between the divisor function of the odd numbers and the "odd" divisor function of power $3$(I don't know if there is a name for function for this type, if there is , sorry for my ignorance), that is  $$\left(\sum_{n=1}^{\infty}\sigma_1(2n-1)x^{2n-1}\right)^2=\sum_{k=1}^{\infty}\left(\sum_{d|n,\frac{n}{d}\mbox{ is odd}}d^3\right)x^k$$ where $\sigma_1(n)$ is the sum of all positive divisors of $n$. This is equivalent to proving $$\left(\sum_{n=1}^{\infty}\sigma_1(2n-1)x^{2n-1}\right)^2=\sum_{k=1}^{\infty}\frac{k^3 x^k}{1-x^{2k}}$$ I tried some calculation via Mathemtica, it seems to be true.This sequence seems to be , but I cannot find some way in cracking this problem. The square seems annoying and I can't get rid of it.  Thanks for your attention and helping hand.
Given the positive integer numbers $n$,prove that $$\sum_{k=1}^{2^{n-1}}\sigma{(2^n-2k+1)}\sigma{(2k-1)}=8^{n-1}$$  where $\sigma(n)$ is defined as $$\sigma{(N)}=\sum_{d|N}d$$  This problem I think is very interesting, and have nice methods?  Thank you everyone.  and I find some of this     or can see:
I'm new to this website. I've checked the other topics and couldn't find the answer to my question. I've also tried to find the result with TI-CAS but the only thing it does is simplify the inequality which really doesn't help at all.  the equation is : $1000n^3 &lt; 2^n$ Now I've been trying to use log but it gets me nowhere (I haven't seen this form before), the answer is supposed to be 24 but I'm not sure. So my question is how to solve step by step this inequality for n. Could you guys help me with this one? User and associated Post Ubuntu on 32-bit UEFI (only) based tablet pc I've created a bootable USB of Ubuntu 16.04 64-bit with Rufus, setting the drive to be a GPT for UEFI. Then I booted my tablet PC and I ran the UEFI BIOS, from which I turned secure boot off. I then restarted and went again on the BIOS to boot from my key, but all I got was a black screen that lasted few seconds and then it brought me back to the BIOS. I read about a bootia32.efi file and other procedures that can be followed, but I didn't find a good explanation of what I need to get done. My tablet PC runs an Intel Core Atom Z3735F and has 2 GB of RAM. If you have the answer please let me know. Is there a law that you can add or multiply to both sides of an equation? How do you remove Nvidia's proprietary drivers? I always seem to have grief with Nvidia drivers and have often had to re-install Ubuntu: is there a better way to remove them? When I first installed the drivers, the display came up on 640 by 480.  I managed to set 2 screens to 1024 by 780 which was usable, but then Ubuntu recommended installing proprietary drivers (nvidia-current-updates 295.40-ubuntu1). After doing this and re-booting it shows LAPTOP and offers only 640 by 480 on a single screen.   Is there any way to revert to the original driver? Is there any way to find out if the new drivers work before installing them? - For years I have been hoping that it would be all fixed in the latest release… I always seem to fall for this! PlotLegends disappearing with list of functions What is the setting in bash for globbing, to control whether * matches dot files I was surprised recently when I did something like mv ./* ../somedirectory and found that files like .gitignore were not moved.  I do most of my work in zsh on OS X, and this surprise bit me in bash on CentOS.  I tried bash on OS X and found the same behavior: * does not match dot files.  This seems very undesirable to me, but apparently it's the bash default.  (It may be the zsh default too for all I remember, but I may have changed it years ago in my .zshrc and forgotten it ever worked differently.)  How can I configure bash to behave as I expected: for * to match all files, and not ignore dot files.  In case this is at all unclear, here's how to reproduce it    cd /tmp mkdir {t,d}est touch test/{.,}{1,2,3,4,5,6,7} ls -hal test mv test/* dest ls -hal test     # notice dot files are still there ls -hal dest     # notice only some files were mv'ed Adding Basemaps from Google or Bing in QGIS ArcGIS Desktop has the option of using basemaps from ArcGIS Online. Does QGIS have any such options? Almost all passports I have seen have a place of birth field on them. For instance, on the US Passport,    Is there a specific reason to this? Firstly, how do you gain the score exactly? I know it shows in the bar on the side of the map, but I remember a match where I was lone-wolfing and still got a pretty high score in that bar. I suppose my other squad members were together, but do I also gain for their work?  Also, does the score reset when I start a new game? I mean, I play support with the Offensive perk and it took me a long game of conquest to just unlock lvl-3 extra grenade. In the next match, that 'nade is gone. Don't I keep it forever? Monotonicity of the sequences $\left(1+\frac1n\right)^n$, $\left(1-\frac1n\right)^n$ and $\left(1+\frac1n\right)^{n+1}$ converting date string to date format Is the electromagnetic spectrum discrete? I'm just starting to learn physics and I have a question (that is probably stupid.)  I learned that energy levels that the bound electron can have are discrete. I also learned that when an electron transitions from one level to another, a photon with a specific wavelength (energy) is created and released into the wild.  My question is this: are there holes in the electromagnetic spectrum, values of frequency that the created photon can't possibly have? Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$ Transaction log reported full but is not Exam with 18pt main font How do I get Mediatek MT7630E 802.11bgn Wi-Fi Adapter working? It worked with the pre-installed Ubuntu, but after re-installing Ubuntu dual-boot with Windows 8.1, the WiFi is not working.  Is there a driver for this adapter that will work on Ubuntu 12.04? I have enabled platform encryption at my DE and selected Case Subject, Description and Comments to be encrypted.  I have a created a sample case with these values:    And when I query from Developer Console it returns as follows:       How do I know, subject and description are encrypted at Rest? Is there any indicator which can confirm that this data is encrypted?   By the way, in case of File/attachment there is isEncrypted attribute is available which denotes the same as follows. When I upload an image to Stack Overflow, I cannot see my JPEG file in my post. It's all blank. I've tried it with different files, but gotten the same result. Many of the various items have a list of enchantments that would be ideal. For instance, I might want Infinity, Power, Flame, and Punch all on a bow... and at the highest levels. Barring incredible luck, I will never get those with an enchantment table.  I can enchant books until I find those spells, and add them to the bow with an anvil. However, at some point (I haven't yet figured out how many times) it will no longer allow you to add enchantments to an item, saying that it's "too expensive". There is a hidden counter that is incremented every time, and once this goes past a threshold you can never add more magic (though, I've read in places you can continue to repair the item at great cost indefinitely... not clear if that's true or not).  This counter is cumulative, such that if you combine two items' magic into a single item, that won't sidestep the limit.  However, I've only just caught on that one can also combine two books. I can take two Power I books, and make a single Power II book. And I can take two of those, and get it up to Power III, and so on, but also have two spells combined in a book.  If I do this until I get all the magic I need onto a single book (or even just two or three books, supposing the books max out), will I get to the point where the books will no longer combine with the item because it's too expensive? Or are books exempt from the cumulative counter thing? It would suck to spend so much time and effort to get a powerful book and discover that it can't actually be used on anything.  Also, is it true that you can indefinitely repair, as long as you have the repair material and are at level 39? Or is that a misunderstanding of the rules (or maybe just an earlier version of the game and it no longer works like that... I'm on 1.8.9)? Does that only work for those items that can be repaired with diamond/iron, or will it work for fishing rods and bows as well?