# #, %O, and so on

I woke up this morning to find someone had added a new button to my basic calculator. Besides the $$+$$, $$-$$, $$\times$$ and $$\div$$ buttons, there's now also a $$\#$$ button. I figured, "Well, this probably won't blow anything up" and set to trying it out.

It didn't blow anything up.

What it did do was give me the following results: $$\begin{array}{rcrcr}2&\#&3&=&2\\ 55&\#&7&=&11\\ 10&\#&5&=&4\\ 7&\#&4&=&0\\ 91&\#&91&=&4\\ 230&\#&1&=&3\\ 1&\#&84&=&15\\ 100&\#&3&=&1\\ 738&\#&3&=&8\\ 5&\#&192&=&14\\ 11&\#&22&=&48734\\ 55&\#&6&=&6\\ 5&\#&48&=&10\\ 3&\#&2&=&1 \end{array}$$Why?

The prose is irrelevant. In particular, ignore the stuff about explosiveness. There's no hint there.

There's a reason the person chose the $$\#$$ symbol for this operation.

• Does the Commutative property hold for operator # (e.g. 2 # 3 = 3 # 2 = 2)? – Conifers Jul 26 '19 at 8:47
• That can be a headstart to solving the problem but I believe it's better left for us to figure out 😅 – Mike Karter Jul 26 '19 at 9:02
• This is an intriguing puzzle which has so far got 9 upvotes and 212 views. But it seems to have defeated us all. Please can we have some more examples of #? Perhaps some that (to someone in the know) reinforce confidence that a certain idea is right, where a single example might only suggest that idea. – Rosie F Jul 27 '19 at 6:38
• @RosieF I've added three more examples and have otherwise edited the question. – msh210 Jul 27 '19 at 20:25
• @Conifers I've added three more examples and have otherwise edited the question. – msh210 Jul 27 '19 at 20:26

Found it, thanks to number 48734

1) Go to https://oeis.org/
2) Search for $$A00000x$$, where $$x$$ is first number. It starts like $$A000000$$, so for $$x=5$$ it would be $$A000005$$ and for $$x=738$$ it would be $$A000738$$ (Or just $$Ax$$ seems to work too)
3) Look for $$y$$th number, where $$y$$ is second number (don't forget it starts sometimes from 0, so look at $$y$$ or $$y+1$$).

• That's one interesting "basic calculator" to either have an internet connection, or have the entirety of OEIS saved on disk ;) – Birjolaxew Jul 28 '19 at 23:12
• @Birjolaxew It may make sense for some, I use Wolfram alpha for "basic calculations", and anything can be a "basic calculator" if you only use it for that. Or, the OP just wanted a basic OEIS surfing machine. – Matthew Roh Jul 29 '19 at 4:46
• +1. So close. It doesn't quite work for all of them. And how does this explain the title of the question? (Hint: The preceding two sentences are related.) – msh210 Jul 29 '19 at 6:06
• @msh210 It does actually work for all of them, with the caveat that you have to use the list view - some of the sequences on OEIS are 0-indexed, some are 1-indexed, so counting manually doesn't always work out. – Birjolaxew Jul 29 '19 at 7:00
• Hmm, the basic calculator should not have ability to query sequence on OEIS... – Conifers Jul 29 '19 at 9:16

The last RHS factors out as

$$48734 = 2 \cdot 7 \cdot 59 \cdot 59$$

But getting two numbers from LHS and multiplying was out of the question because of examples like

$$\begin{array}{rcrcr}7&\#&4&=&0\\ 91&\#&91&=&4 \end{array}$$

So let there be

$$\begin{array}{rcrcr}A&\#&B&= &(a \cdot A + b) + (d \cdot B + e) \end{array}$$

What next...

• Someone please edit this answer to better formatting. I'm new to Mathjax thing. Help here and suggestions for the future will be appreciated. Thank you. – Mike Karter Jul 26 '19 at 8:54
• For future reference there is MathJax basic tutorial and quick reference. – Weather Vane Jul 26 '19 at 8:58
• how do you spoiler-tag mathjax? – Somebody Jul 26 '19 at 16:18
• @Somebody it can be frustratingly tricky to get the spoiler tags right. Here, I merged each Mathjax statement within \$ to a single line. After the first non-spoiler (visible) lines, begin the spoiler text with no empty line. On each line I added the spoiler tag >!  (with one space) at the front, and a double-space at the end (including empty lines) which forces a line break. If you add an image, just add the spoiler tag afterwards in front and the double space at the back. The easiest way to figure it out, is to "edit" another question or answer and have a look – but then "cancel"! – Weather Vane Jul 26 '19 at 17:28
• (... so how does this contribute toward a solution? As this stands, it's Not an Answer, not even a partial one. Having fragmentary thoughts on aspects of a puzzle might be comment-worthy, but you probably want at least a germ of an idea that seems to lead forward before you should post as even a partial answer.) – Deusovi Jul 26 '19 at 18:50