#of divisors Series

[oreotheory]

Looks like 578 is the first number that goes to infinity!

That's really interesting! I had imagined that every number would end in a loop (of course who knows maybe this one does eventually). The post of @henryzz using powers of prime seemed to at least superficially confirm this impression.
(Though I like @Villiam Furik's idea of making the behavior of primes less predictable, I think that sequence isn't as exciting because the end term always ends up being a factor or multiple of the original number, whereas the original one I proposed is often random and unpredictable. (Furik Conjecture for primes: p*2/1=2p))
I'd be curious to see a tree for the original series, to see what numbers connect to which. Also very curious which numbers go to infinity. By the way, @Stargate38's 10^32-1 is a multiple of 578 (unless google's calculator is fooling me...)

[Batalov]

Originally Posted by oreotheory

By the way, @Stargate38's 10^32-1 is a multiple of 578 (unless google's calculator is fooling me...)

Apparently it does.
How can an odd number (10³²-1 is odd) be a multiple of an even number (578 is even)?

[Viliam Furik]

Originally Posted by oreotheory

(Though I like @Villiam Furik's idea of making the behavior of primes less predictable, I think that sequence isn't as exciting because the end term always ends up being a factor or multiple of the original number, whereas the original one I proposed is often random and unpredictable. (Furik Conjecture for primes: p*2/1=2p))

The conjecture originally was that these prime loops (p -> 2p -> 2p) are the only loops and that all numbers terminate in such a loop. @henryzz pointed out more formats (actually a pattern for infinitely many patterns of infinitely many loops), extending the conjecture.

I will try to whip up a proof of this, as it seems to me to be obvious (that for every starting number it will eventually get into a loop of that format (or some another unknown format), and none will go forever). If I get somewhere, I will post an update on this.

[oreotheory]

Originally Posted by Viliam Furik

The conjecture originally was that these prime loops (p -> 2p -> 2p) are the only loops and that all numbers terminate in such a loop. @henryzz pointed out more formats (actually a pattern for infinitely many patterns of infinitely many loops), extending the conjecture.

I will try to whip up a proof of this, as it seems to me to be obvious (that for every starting number it will eventually get into a loop of that format (or some another unknown format), and none will go forever). If I get somewhere, I will post an update on this.

Oh, got it now. It's been true for every number I've tried in the pattern. Also not sure how to prove it. Squares and primes seem to be the building blocks, but different squares and primes will act differently...

[bur]

I knew there'd be a simple in-house solution... thanks.

[bur]

I somehow messed up my post, it was supposed to quote Batalov's elegant tau(n) = #divisors(n).

Some variations: Instead of tau, one could use omega, bigomega or even sigma.

Or instead of division/multiplication, use subtraction/addition. Which to use could be determined by e.g. µ. To prevent a quick stop at µ = 0 one could do a(n) = a(n-1) + [c+d*µ] * tau(a(n-1)). The choice of c,d influences the growth or shrink of the series.

Or a(n) = a(n-1) + (-1)^omega(a(n-1)) * phi(a(n-1)). I toyed with that function for a while. Most even numbers with additional odd prime factors grow to infinity, containing ever larger powers of 2. Odd numbers are more interesting. The script below computes the sequence and prints the factors. Again first parameter is n0, second parameter the number of terms to calculate. If omitted, it runs until a prime is reached and a(n) = 1.



omegaPhi(n) = {
return(n + (-1)^omega(n) * eulerphi(n))
}


omPhiSeq(n,j) = {
i = 0;
until( n == 1 || i == j,
print(i,": ",n," = ",factor(n));
n = omegaPhi(n);
i++
)
}


Sorry for hijacking the thread, but I thought it was a similar idea. :)

[oreotheory]

The idea is very cool. Forgive me, but I have zero knowledge of coding. Could you explain to me (like I'm four) where in the code I type in the input number? I've been trying to do so on this online pari executor with no luck: https://www.tutorialspoint.com/execute_pari_online.php

[Viliam Furik]

Originally Posted by Batalov

Apparently it does.
How can an odd number (10³²-1 is odd) be a multiple of an even number (578 is even)?

It must be a rational multiple

[bur]

Just copy&paste the code into GP. Then you can run it via omPhiSeq(n,j) where n is the first value of the sequence and j (which is optional) is the number of terms you want to calculate. For example omPhiSeq(63,3) calculated the first 3 terms, whereas omPhiSeq(63) calculates terms until a prime is reached.

Here, we'd enter a loop since for prime p: eulerphi(p) is p-1, omega(p) is 1, hence: p - 1 * (p - 1) = 1. Followed by 1 + 1 * 1 = 2, followed by 2 - 1 * 1 = 1. Thus, reaching a prime is a natural stop for the sequence.

If you want to modify something, this is the line you want to edit: return(n + (-1)^omega(n) * eulerphi(n)). Substitute omega by bigomega or sigma or whatever. You can also a constant, such as return(n + (-1)^omega(n) * eulerphi(n) + 3) which'll add 3 to each term. No idea what'll happen then. ;)

Very briefly spoken, name(a) means that you run command name with the parameter a.


edit: By GP I meant the PARI/GP software if you're not familiar with that. Either download the standalone or try the browser version.

[Viliam Furik]

Originally Posted by Viliam Furik

9 -> 18 -> 3 -> 6 -> 6

I've noticed a mistake I made. Oddly enough, it fixed itself. Under my gcd rule, 9 goes to 3, not 18 (while doing those first few numbers in my head, I probably forgot to add 1 to the power 2 in 3^2), but 18 does indeed go to 3.

[Viliam Furik]

While searching for information about the number of factors of a number and the properties of a function of the number of divisors, I've found something which I knew about, but forgot. https://en.wikipedia.org/wiki/Refactorable_number - Curtis Cooper researched them. Those are precisely the numbers for which the first-definition sequence decreases.