Extensions to drivers/guides

henryzz · 2013-09-16, 18:50

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 · 2013-09-16, 21:29

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.

2013-09-16, 18:50	#1
henryzz Just call me Henry "David" Sep 2007 Cambridge (GMT/BST) 2×2,909 Posts	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: 2^53^27: 3^2 and 7 are both maintained by 2^5. Power of 2: 3 2^53^2713: 13 can be added as it is kept by 3^2. Power of 2: 4 2^53^27^213: It is possible to add another power of 7 as 13 will keep that along with 2^5. Power of 2: 1 2^53^27^21319: sigma(7^2)=57=319 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 2^53^257^21319: 5 can be added because of 19. Power of 2: 4 Another example: 2^20712733713^236131: Power of 2: 18 2^207^219512733713^236131: Power of 2: 18 2^207^21951273373^213^3: Power of 2: 13 2^207^21951273373^213^317: 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 2^33^25137 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 2^33^25137p is 1-2/34/56/712/13=263/455=57.8% This is not very good when you compare it to a driver like 23 which has 33% or 2^27 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, 2^207^21951273373^213^317p above has a 18.6% chance of raised powers.

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2013-09-16, 21:29	#2
henryzz Just call me Henry "David" Sep 2007 Cambridge (GMT/BST) 13272₈ Posts	2^73^451711^2719p has a high enough power of 2. 53.7% chance of raised factors. 2^73^35^23117p has a high enough power of 2. 51.4% chance of raised factors. 2^877337195p has a high enough power of 2. 37.7% chance of raised factors. 2^14731151195p 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.