TITLE: What problem would you base your mathcoin on?
QUESTION [36 upvotes]: Recently, a variant of electronic currency, based on prime sextuplets,
broke the record in generating the largest known set of six primes, packed as closely as possible, that is, a sextuple $(p,p+4,p+6,p+10,p+12,p+16)$
such that all numbers are prime.
I am not certain how significant this is to the mathematical community, 
but there are always a need for more data, which can be used to formulate conjectures. 
So, if the mathematical community had access to a large amount of computer power, what would be worthwhile compute, and why? Ignoring details about how to make the computation run in parallel and all that.  
Personally, I would like to see the Ehrhart series for the Birkhoff polytopes, these are only known up to $n=10$.
Another series that would be nice to know some more entries of are the number of 1324-avoiding permutations in $S_n$ for larger $n$, (record from 2013 2014 is for $n=36$). This is the smallest instance of enumerating pattern avoiding permutations where no formula is known.
Finding more Mersenne primes is also quite interesting. I wonder if I will see the 100th Mersenne prime in my lifetime.

REPLY [9 votes]: Apologies if this answer is nitpicky and diverges slightly from the questions intent, but this particular soft question appears to warrant a soft answer. 
Implicit in your term "mathcoin" are assumptions that the problem:  

Allows new solutions to be found in regular tunable time intervals,  
Is memory hard so that specialized ASICs cannot speed up the computations, and 
Any significantly faster algorithms represent the sort of dramatic advancement that it'd emerge from across the mathematical community simultaneously and not from a secret project. 

Of course, collisions in cryptographic hashed satisfies 1 and 3, while a memory hard cryptographic hash like scrypt or argon2 satisfies 2 as well.
Almost all the answers proposed here thus far fail even 1 and maybe all here fail 2.  Worse, there are probably no calculations that both pass 3 and represent useful mathematical (resp. scientific) research, at least not for 2 (resp. 10) years.  
In short, a naive "mathcoin" as suggested here sounds impossible to do correctly.  There might be less naive approaches where perhaps any problem goes and the problem's difficulty is established by previous attempts, but that's actually solving a social problem to achieve 1 with widely varied math problems as simply a source of randomness that defeats ASICs.
As an aside, there are proof-of-useful-work systems that could possibly support a bitcoin like currency without wasting resources, but again they all provide useful social services like file storage (filecoin) or anonymity via onion routing. 
Update 2018:  There is still no evidence of any suitable mathcoin problem but there are seemingly suitable science and engineering problems.  Examples:
Proof-of-moon:  All miners must buy telescope arrays to explore the night sky looking for specific astronomical events.  If you find one, then reporting it in a canonical way and hashing the report with a nonce can win a block, with the difficulty possibly adjusted by the event.
Proof-of-vulnerability:  All miners run advanced static analysis tools on software to discover vulnerabilities in published software.  A miner wins the block with the hash specific characteristics of a vulnerability plus a nonce, so once a vulnerability is found the miner will eventually win a block.  There are several challenges though:  First, you must prevent a vulnerability from being used twice, potentially this could be done by resolving to a particular point in the code, but then projects with more releases become better targets.  Second, static analysis tools have false positives, so you might need to test consequences somehow.  Third, you'll likely need some sort of proof-of-stake system to permit humans to ultimately arbitrate the vulnerabilities.  Forth, you want to tackle closed source software too, so the definition of published becomes tricky.  Fifth, developers might get annoyed with vulnerabilities being posted to some blockchain without even attempting to warn them first.