20220613, 00:49  #12  
"NT"
May 2022
U.S.
19 Posts 
Quote:
(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 10^321 is a multiple of 578 (unless google's calculator is fooling me...) 

20220613, 01:35  #13 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
3^{3}×367 Posts 

20220613, 20:53  #14  
"Viliam Furík"
Jul 2018
Martin, Slovakia
5^{2}×31 Posts 
Quote:
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. 

20220613, 21:05  #15  
"NT"
May 2022
U.S.
23_{8} Posts 
Quote:


20220614, 10:34  #16 
Aug 2020
79*6581e4;3*2539e3
586_{10} Posts 
I knew there'd be a simple inhouse solution... thanks.

20220615, 15:50  #17 
Aug 2020
79*6581e4;3*2539e3
2×293 Posts 
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(n1) + [c+d*µ] * tau(a(n1)). The choice of c,d influences the growth or shrink of the series. Or a(n) = a(n1) + (1)^omega(a(n1)) * phi(a(n1)). 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. :) Last fiddled with by bur on 20220615 at 16:06 Reason: Fixed script 
20220615, 19:05  #18 
"NT"
May 2022
U.S.
19 Posts 
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

20220619, 20:47  #19 
"Viliam Furík"
Jul 2018
Martin, Slovakia
5^{2}×31 Posts 

20220620, 13:14  #20 
Aug 2020
79*6581e4;3*2539e3
1001001010_{2} Posts 
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 p1, 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. Last fiddled with by bur on 20220620 at 13:17 
20220621, 16:12  #21 
"Viliam Furík"
Jul 2018
Martin, Slovakia
5^{2}×31 Posts 

20220628, 14:46  #22 
"Viliam Furík"
Jul 2018
Martin, Slovakia
5^{2}·31 Posts 
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 firstdefinition sequence decreases.

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