You would still get sha256 hashes from mining blocks, but very particular ones, as you would get 9 out of the sum of its digits, in base 10, and of a few powers of them, as well.
It would be a bit like finding new digits of the number PI. Why do it? For research.  
It may prove useful in the future.
I think this would not be better than finding new digits of PI but also not worse. It is at least something you can show other people as an achievement.
Pretty much the same reason athletes and so, keep doing all sorts of efforts to beat records. Human achievement, curiosity about our limits, research...
This might not convince you but there you go, here is my answer.
We could then say that bitcoin is backed by something and has intrinsic value: it would be finding rare and precious numbers.
And in the end, it might even prove to be useful knowing such numbers. It's better than nothing, at least, I think.

Here is a story about a number:
~ The Hardy–Ramanujan number: 1729

This number got its start when British mathematician G. H. Hardy visited Indian mathematician Srinivasa Ramanujan while Ramanujan was ill in the hospital. Hardy later recalled:

"I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. ‘No,’ [Ramanujan] replied, ‘it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.’ "

The two cubes are 1729 = 1^3 + 12^3 = 9^3 + 10^3. Numbers that are the smallest number expressible as the sum of two positive cubes in n distinct ways are called “taxicab numbers” for just this reason.

0x0000000000000000055555555555555555555555555555555555555555555555^17=

9567639692108043072811450028127400518281204094818345885318236555087134713804395 7298464973764356275103554879331415138053356424891863760353039671546154304315081 1566469523986070256935592964043364944268216023279526986939181898116614435967889 3616638606887122504304800352366090111525854262617481678198448227772673263250855 8000685531324347333260410651563340863148349582073424253019854463763099032912118 7652471312435535069526587697537625186126548091442969484731037914577069735859518 1412394471985250667465713838607113372333720574309575798121489470388950618234180 5940563991344157394076447656598555239895453208891176505721576013557444261355363 5928628909184421289989071553328104314106992525434339700388013380688641778490970 9293592938336469293401095932512404377328638497665080639978453085011547878346899 6745955324447707973205619438293306935554825314723008561120809590470039092678033 0189838168617004322658962332322688147473727239414890354831795951410908216094970 703125

Finding numbers this large with such described particularities - every 10 minutes - wouldn't be some impressive achievement?...

You could still impose that the hashes and their powered values are lower than some target values, so it could be made to have blocks every 10 minutes also.

I'm just saying that would be slightly more interesting to be able to know about the distribution of these particular numbers. Nothing really impressive, for sure, but I think worth thinking and perhaps the spark for something better and more sound (and more like primecoin).

Yes, I think this would require to wait for a hard fork (with some other more important changes, possibly) because you could not use old clients anymore if this PoW was used. Certainly, ground for an altcoin, indeed. But who knows...

I think it is pretty much the same math.
When the miners hash the block - A - with a given nonce there is:
10% chance it will have one leading zero.
There is also 10% chance that the sum of all digits of the decimal representation of that block hash is 9.

The chance for two leading zeros is 1%
For the sum of all digits of A and of A^2 to be nine we also have 1% chance
and so on.

For each leading zero or next level of casting out nines we get 10 times less chance.
It is exponential in exactly the same way.

So if my math is correct, at present difficulty problem, we would be finding,
A such that:
casting out nines of A is 9
and
casting out nines of A^2 is 9
and
casting out nines of A^3 is 9
and
casting out nines of A^4 is 9
and
casting out nines of A^5 is 9
and
casting out nines of A^6 is 9
and
casting out nines of A^7 is 9
and
casting out nines of A^8 is 9
and
casting out nines of A^9 is 9
and
casting out nines of A^10 is 9
and
casting out nines of A^11 is 9
and
casting out nines of A^12 is 9
and
casting out nines of A^13 is 9
and
casting out nines of A^14 is 9
and
casting out nines of A^15 is 9
and
casting out nines of A^16 is 9
and
casting out nines of A^17 is 9

every 10 minutes...

The probability of finding it with only one nonce being tried out is:

0.000000000000001%
=1E-15%, in scientific notation.

In both cases.
(assuming the distribution of numbers divisible by 9 is normal)

It would be interesting if PoW could be used to discover proprieties of numbers.
For that matter I propose this variation to the PoW method:
Instead of looking for a certain given number of leading zeros in the next hash, why not be looking for consecutive casting of nines in the hashes and their powers of 2,3,4,...
For a casting of nines of a number to be zero the sum of all the digits of that number has to be nine.
So, according to the difficulty set, we would be looking for hashes and their powers of 2,3,4,5,6... so that for each its digits added up to nine.
So miners would get a hash, and if the sum of all digits of its decimal representation was nine and so the cating of nines gave zero, the power of two of the hash would be calculated and if its casting of nines was also zero the power of three and its casting of nines would be calculated, and so on for power of 4, and so on, according to the difficulty set.
For instance, the difficulty could be saying the casting of nines of the hash has to be zero and the same for its square; its cube and so on till the power of 20. When the difficulty needed to increase we would go for power of 21 and further... Always adding more to the number of powers that also had its decimal representation's sum of digits the same.
Maybe more interesting number proprieties could be found but this is the best I can think of.
Imagine that for a given difficulty we had the hash and its powers of up to 20 all so that the sum of its digits was nine and so having a casting of nines of zero. As the difficulty increased we would find more improbable and interesting numbers with a specific propriety.

This.

+1

I have just watched it and think it was damn good.  
I think it is a quite good introduction to bitcoin.
The public didn't laugh much when Gavin said there was no GavinCoin but I found that funny  

Why not asking the author of the photo?

If this wasn't enough prepare... Sit down and have a deep breath...

Ready?!

When bitcoin lauched I was in Italy, on hollidays.
I had spare time and wrote an article about currencies with a very trippy and visionary banknote image.
See the blog post here:
http://velocidadedaresposta.blogs.sapo.pt/7375.html

Look at the date of that blog post, my username handle, and tell me how odd?
I'm not Satoshi but have been using bicomplex and bichoverde for ages.

And Ihave a very odd patent about digital money tokens that was registered before bitcoin.

So I guess I make a really good case for being close the source...

So, now, can I get your attention when I say I know who is Satoshi?

But there is more. This gets really interesting. I used to send him emails to antonio.f.alexandre@clix.pt. I know as a matter of fact that he received my emails although he never replyed to them. He once told my mum that he used to receive my emails. Some interesting ones and some quite crazy. But the really interesting stuff is that one of my last emails to him was a link to a blog post that was talking about some p2p money that was out there. This was in 2009!!!
Unfortunately I dont have the link of that blog post but would be really interesting for me to prove that I got to know about p2p money back in 2009 when I was studying Computer Science and finding it sufficently interesting to send an email to my dear uncle forwarding him the link of that blog post where p2p money was being discussed.
After that, surprisingly enough, the emails I sent him would always bounce back so I gave up sending him emails or trying to contact him ever since.
But I only entered bitcoin world in August 2011 when Ray Kurzweil posted about it on the AI-news mailing-list.

See the email address mentioned at:
http://cienciapt.net/pt/index.php?option=com_content&task=view&id=81114&Itemid=298
Goos luck in trying contacting him and keeping us posted.

Believe it or not. It is true.

Quite so; I think I was ignited by that.

The poll was just to make the topic more interesting and engaging; I knew it wouldn't be telling much.

I don't believe for one second in the Nick Szabo being Satoshi theory, indeed.
But bitcoin name was certainly inspired by 'bit gold' name I reckon. It is a bit of 'digicash' and a bit of 'bitgold' name, I guess, with the subtle irony that the ledger makes this new form of money 'thingness' very different from the physical and tangible coins we are used to.

Once my uncle, I think this was around 1997, was telling me:
"The real thing is focusing on something that really prevails everyone's affairs.", and he asked, "what do you think is that one think you could be studying that affects everyone's life?"
I was a young man studying Electrical Engineering at that time, and I answered promptly, a bit proud of myself, "Electricity, of course."
Another time he asked me what was the one things that I would be able to teach our ancients if I could go back in time. What would be the revealing and enlightening things I could bring there with me, with the knowledge I got from our time.

These are the kind of little things that make me think my uncle is Satoshi. And the more I think of it the more it makes sense. It makes it all look much more than plausible, it simply leaves me without any doubt that could only been him.

I know this will sound really delusional; but one day I am sure you will all acknowledge that I was right, and that my certainty was not as trivial as it may seem.

Yes, you are right and it makes perfect sense to think he 'ehvedadooanonimato' is brazilian, from São Paulo, but then again if he is the Satoshi I have in mind, quite interested in world affairs and stuff, wouldn't be hard to believe he reads publications like "Revista Veja" magazine.
Back in that time, he would only need to read an article like http://veja.abril.com.br/noticia/economia/nota-fiscal-paulistana-comeca-a-valer-hoje/
to get to grips with convincingly seem Brazilian.
I get your point, but you have to consider all angles since we are talking of such an elusive man as Satoshi.

Yes, he is definitely quite knowledgeable about linguistics, modern formal logic and stuff.
Among many other things like Ethics, Economics and world affairs.
A bit like Noam Chomsky, I would say, but more of a mathematician, able to go deep into Modern Algebra and stuff.
He doesn't have made any contributions to cryptography but he certainly can go much deeper into it than 99.9% of the people can.
This is a guy with an IQ of 195, that studied Mathematics at Harvard in the 60's.
There is no way he doesn't understand 99% of Satoshi's whitepaper. That's simply impossible to believe, for me, at least.

I really have no doubts he is Satoshi and I know that time will prove me right.

A kind of Richard Feynman with a level of knowledge that defeats explanation.
A genius. A very elegant and intelligent man with a very tender sense of humour.
Definitely an extremely high IQ, well versed in the mastery of languages,
(Portuguese, Spanish, French, Italian, English, but also a bit of German and Hebraic...)
Found of the Japanese culture; a libertarian and progessist; a very modern intellectual.
A very sensible man, with a very centered heart.
A bright mathematician that finds Fermat's theorem and its proof fascinating; that finds gold, water, sun and moon fascinating. In a word, Satoshi.
The kind of intelligence that would learn the fundamentals of C++, to the level of knowledge of Bjarne Stroustrup, the creator of C++, in a weekend or two...
A guy that is always surprising you by how much he knows, and that will know that it is Roberto Benigni you mean, and not Noam Chomsky or William Gibson or John Searl or Bertrand Russell or John Travolta or Donald Trump or Rupert Murdoch or Warren Buffet. Who else? Satoshi.
Literally, there is nothing you can throw at him that he doesn't already know. The real dad... Uncle Sam!

A fascinating man.

Wait, he also has a few posts in French.

Here is one of them:
https://bitcointalk.org/index.php?topic=36147.msg480053#msg480053

Yes, but can you indicate us any other user that fits so much Satoshi profile as this? I would like to see which other users you can come up with that matched a Satoshi profile as much as this one...

There is certainly a lot of charm in being kept a secret mystery but the fragrance of a living Master like him is definitely not to be missed, IMHO.
There is so much in a genius like him that is not to be missed...
For me it would be like missing a great spiritual teacher like Jesus or Buddha.
Or missing a great genius like Albert Einstein or Richard Feynman...
I can't help but think the world would be better with this mystery solved and his fragrance being appreciated by us all.

Meanwhile, it's not easy for me to have this position.
It's a tough one, for sure.
But I don't regret it. I'm proud of my position in revealing him.

Maybe I should put this link on reddit but here it goes, I hope you find it interesting:
http://athousandnations.com/2015/04/03/the-nakamoto-consensus

Do you really believe that a guy with his knowledge of computer systems security, that seems to be mining with GPUs since 2010 couldn't code in C++? I don't. That statement seems to me just to outwit us.
He also somehow pretends to be Brazilian, but I'm pretty sure he is our Portuguese Satoshi, well versed in all the languages subtleties.
The way he pretends to be Brazilian and not Portuguese is very clear to me.
I think he wanted to browse a lot the Portuguese subsection but thought he would be more disguised if he passed as being Brazilian.