## 28 October 2007

### PIZZAQ: find a value of the Squuzuu Function

Click image for bigger.

How hard can this be?

In increasing order, the positive whole integers are n.

Each n returns an integer with that funny symbol, which is called The Squuzuu Function. They increase, too.

I didn't just make up the Squuzuu Function. It's not random or arbitary. It's a well-known, even (among certain nerd circles) a famous relationship.

Here's a hint. To define the Squuzuu Function, you don't need the whole table. You don't need to know what numbers went before 8 and 22 to determine that

if n = 8,
then Squuzuu(8) = 22
.

Once you understand the relationship, I could ask

If n = 33,
what is Squuzuu(33) ?

and you could calculate or compute Squuzuu(33) = 10143 without using any previous part of the table, or without having the table at all. In other words, once you understand the relationship, you can make your own table from scratch, or just any one line of the table, with pencil, paper and patience, and then when things get really big, with a fairly simple computer program.

if n = 73,
what is Squuzuu(73) ?

and (with or without a computer's help) you could compute the value of Squuzuu(73) .

In fact, that's what I'll award 1 whole Pizza, with extra cheese, shallots, endives, and shitaki mushrooms for. Given this partial table,

if n = 73,
what is Squuzuu(73) ?

You're on the Vleeptron PizzaQ Honor System, no Googling, no phoning Klaas in Rotterdam. But I think I've done a pretty good job of insulating this from idle cheating.

If anybody sends in a guess, I'm going to be really sorry, because I don't know the answer yet, so I'll have to grind it out, and that should take a few aspirin.

abbas said...

well looking at the first ten numbers, all i can deduce off the top of my head without a pencil and paper is that it's a parabolic function. will dig up a bit more when i have some sleep in me.

Vleeptron Dude said...

Hmmm, when i got your answer i was just about to fall into a coma, too, and then my pewter started to misbehave.

If I were in your shoes, I'd sketch out some of these x,y points on graph paper, and I think it would pretty quick and clearly indicate that it's not a parabola. Anyway, it's not a parabola.

abbas said...

okay so here's the thing. it's the partition function. all the partitions of a natural number, where the natural number is n. which technically means, partitions of n is basically how many different sums can you write that sum to n. but only works for positive terms as noted by bob. initially i thought it was a variation of some fibonacci sequence but that was untrue. though it differs across disciples in british and american, the function can be different depending on whether they agree that 0 is a natural number or not. in this case it is. you can read more on the function here. in any case, the answer to squuzuu(73) is 6185689. and here is someone really bored.

abbas said...

oh and to explain that in lay man's terms.

0 is 0, therefore 1.

1 is 1+0, therefore 1.

2 is 1+1, 2+0, therefore 2

3 is 0+3, 1+2, and 1+1+1, therefore 3

4 is 0+4, 1+3, 2+2, 1+1+1+1 and 2+1+1

5 is 5+0, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, and 1+1+1+1+1

and so on.

Vleeptron Dude said...

Hey! Usually when I have a math(s) PizzaQ, you gripe and complain that it bores and annoys you. Was this one drenched in raspberry chocolate?

I credit 1 large Fancy Pizza to your account, payable the next time we're both in the same place. Which isn't at all unlikely. Every summer we usually make a schlep up to one of the big theater festivals near Toronto. And you made a driveby near my place last year.

AT&T (the phone company ... well, they used to own Bell Labs, a real Deep Thought kinda place) has a huge catalog of integer sequences, and this one, the Partition Function, is Integer Sequence Catalog Number A000041 .

(Fibonacci is Integer Sequence Catalog Number A000045 .)

Like I said. in certain *very nerdy* circles, the Partition Function is rather famous.

But I think realistically, in the trailer-park circles I hang in, the PF is one Very Obscure, rare-ish sort of thing. Not quite as well known as Britney Spears or Tom Cruise.

I went thru a pretty fancy math education and never stumbled across this. It's sort of sub-interesting -- the concept of all the ways you can express an integer in sums of smaller integers just seems ultra-trivial and unimportant, if not downright boring. A voice from deep inside our Inner Schoolboy screams: Who cares?

But in that Ramanujan - Mock Theta Function lecture I went to at Amherst College the other night, the Partition Function turns out to be what Ramanujan's Mock Theta Functions are all about -- and in turn, the MTFs now appear to have deep results that describe such Real-World (yes! Real World! Things you can touch and crash your bicycle into!) behavior as String/Superstring Theory.

As soon as I can find some references to the important things and breakthroughs that are causing so much excitement in this Partition/MTF field which are written in something resembling English, I'll have more posts about them.

HOW THE HELL DID YOU SOLVE THIS PIZZAQ? Do you have a Partition Function sitting around in your garage? Where the heck did you get Number Theory shoved into your skull?

abbas said...

heh number theory is neat. i guess i never mentioned i'm a physics major. which basically means a lot of math and computers and some physics.