Go Back > Factoring Projects > Aliquot Sequences

Thread Tools
Old 2013-09-16, 18:50   #1
Just call me Henry
henryzz's Avatar
Sep 2007
Cambridge (GMT/BST)

2×2,909 Posts
Default Extensions to drivers/guides

I have 3^2 * 13 as factors of one of my sequences currently and it occurred to my the the 13 had stuck around because of the 3^2.
I have been thinking about whether there are any groups of factors that would behave like a driver in that you would need to raise the power of a prime to escape this group of factors.
I have been attempting to construct a good group but haven't succeeded yet as the power of 2 generated by the primes has been too small to maintain the power of 2 in the driver/guide(later in writing this post I succeeded).
Here are a few of my attempts:
2^5*3^2*7: 3^2 and 7 are both maintained by 2^5. Power of 2: 3
2^5*3^2*7*13: 13 can be added as it is kept by 3^2. Power of 2: 4
2^5*3^2*7^2*13: It is possible to add another power of 7 as 13 will keep that along with 2^5. Power of 2: 1
2^5*3^2*7^2*13*19: sigma(7^2)=57=3*19 so I could add 19 into the mix. Adding another 3 would mean we don't keep 13. The power of 3 can't rise or fall now though. Power of 2: 3
2^5*3^2*5*7^2*13*19: 5 can be added because of 19. Power of 2: 4

Another example:
2^20*7*127*337*13^2*3*61*31: Power of 2: 18
2^20*7^2*19*5*127*337*13^2*3*61*31: Power of 2: 18
2^20*7^2*19*5*127*337*3^2*13^3: Power of 2: 13
2^20*7^2*19*5*127*337*3^2*13^3*17: This one is particularly stable as the power of 3,5,7 can't change. The problem is still the power of 2 being too low. Power of 2: 14

While writing this I discovered
2^3*3^2*5*13*7 is relatively stable. The power of 2 is fixed although the powers of 3,5,7,13 can all change upward which starts things crumbling.

Basically we need to find a group of factors that protects each other well. The probability of raising a power in 2^3*3^2*5*13*7*p is 1-2/3*4/5*6/7*12/13=263/455=57.8% This is not very good when you compare it to a driver like 2*3 which has 33% or 2^2*7 which has 14.3%. What is the best we can get? Can we come close to or beat some of the drivers? Apart from powers of 2, 2^20*7^2*19*5*127*337*3^2*13^3*17*p above has a 18.6% chance of raised powers.
henryzz is offline   Reply With Quote
Old 2013-09-16, 21:29   #2
Just call me Henry
henryzz's Avatar
Sep 2007
Cambridge (GMT/BST)

132728 Posts

2^7*3^4*5*17*11^2*7*19*p has a high enough power of 2. 53.7% chance of raised factors.
2^7*3^3*5^2*31*17*p has a high enough power of 2. 51.4% chance of raised factors.
2^8*7*73*37*19*5*p has a high enough power of 2. 37.7% chance of raised factors.
2^14*7*31*151*19*5*p has a high enough power of 2. 37.5% chance of raised factors.

2^22 and 2^26 look promising assuming someone can get them to work.
henryzz is offline   Reply With Quote

Thread Tools

Similar Threads
Thread Thread Starter Forum Replies Last Post
Extensions Fred PrimeNet 5 2019-04-13 16:27
Finite field extensions carpetpool Abstract Algebra & Algebraic Number Theory 3 2017-11-15 14:37
Extensions to Cunningham tables Raman Cunningham Tables 87 2012-11-14 11:24
Extensions R.D. Silverman Cunningham Tables 4 2011-04-27 09:27
DC Project Guides OmbooHankvald Information & Answers 21 2006-01-24 18:44

All times are UTC. The time now is 03:28.

Mon Mar 1 03:28:19 UTC 2021 up 87 days, 23:39, 0 users, load averages: 1.16, 1.72, 1.84

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2021, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.