Open-ended odd sequences

[gd_barnes]

I haven't seen this mentioned anywhere here. Below are all 11 open-ended odd sequences <= 5^6 and a small snippet about most. Of course they all merge with an already-known open-ended even sequence but just the few number of them makes them interesing.


3025 = 5^2*11^2; merges with 1074; a perfect square and the lowest odd open-ended sequence
7225 = 5^2*17^2; merges with 1134; a 2nd perfect square
8015 = 5*7*229; merges with 1074; the lowest that is not a perfect square
8427 = 3*53^2; merges with 1074; the lowest not divisible by 5
12095 = 5*41*59; merges with 1074; the lowest without a squared factor
12475 = 5^2*499; merges with 1074
13225 = 5^2*23^2; merges with 2982; notice the pattern of 55^2, 85^2, and 115^2 as the 3 lowest sequences that are a perfect power of any kind.
14641 = 11^4; merges with 1464; the lowest not divisible by 3 or 5 and yet another perfect square.
15095 = 5*3019; merges with 1074
15407 = 7*31*71; merges with 1074
15625 = 5^6; merges with 3906; the 5th perfect square

5 perfect squares out of 11.
7 out of the 11 merge with 1074.

Interestingly many of these have an i=1 value of, you guessed it, 3025, the lowest sequence, which subsequently quickly merges with 1074. It's the fact that their 1st index value is 3025 despite multiple different starting factorizations that makes it unusual and interesting.

I came up with this by running a batch process on aliqueit.exe, turning off the stop on merge indicator, and manually stopping the program whenever I saw it get get past an index of 300 and checking for the sequence being open in the DB.

This begs the question: What is the lowest open-ended odd sequence that merges with an even open-ended sequence that has not yet been searched? Hum...it would likely have to be > 300K and more than likely > 1M.

Other large open odd sequences likely include 3^10, 3^14, 3^18, 3^20, 5^8, and 5^12. According to the DB, 3^18, 3^20, and 5^12 all merge with with an even sequence > 40M. 5^12 = 244140625 merges with 41561044. Assuming that no one has searched 41561044, that one would certainly qualify although I'm sure it's not close to one of the smallest ones.

Is there other information on odd sequences or are they ignored because they always merge (I think) with a lower even sequence?


Gary

[Greebley]

8015 = 5*7*229; merges with 1074; the lowest that is not a perfect square
12095 = 5*41*59; merges with 1074; the lowest without a squared factor

The above seems contradictory.

There is also the question of what odd open-ended sequence doesn't merge with another sequence. I verfied the number is over 5 million but I don't know what it is.

[gd_barnes]

Originally Posted by Greebley

8015 = 5*7*229; merges with 1074; the lowest that is not a perfect square
12095 = 5*41*59; merges with 1074; the lowest without a squared factor

The above seems contradictory.

Oops. You are correct. 8015 is the lowest that is not a perfect square and is the lowest without a factor that is squared. The 2nd quality there is no longer so unique after all.

Originally Posted by Greebley

There is also the question of what odd open-ended sequence doesn't merge with another sequence. I verfied the number is over 5 million but I don't know what it is.

Oh wow; now THAT is a very interesting question.

I had ignored that possibility because I didn't think that it was mathematically possible but I just now convinced myself that it is. I thought odd sequences ALWAYS had to drop on index=1 because they had no factor of 2. How wrong I was. If they dropped, then by definition, they would merge with a lower sequence.

So the next question is: What is the lowest odd sequence that drops on index=1? I messed around manually with the D.B. for a little bit and found sequence 3*3*3*5*7=945, which has a value of 975 at i=1.

This is a whole other research area on these sequences. Here are the possibilities:
1. As you have reasearched, what is the lowest open-ended odd sequence with with no merges?
2. What is the lowest odd sequence that has a higher i=1? That will be easy to find so I'll try it.
3. What is the lowest odd sequence that has a higher i=1 followed by a higher i=2?
4. Continue on #3 for consecutive increases thru i=3, i=4, etc.

Edit: 945 is indeed the lowest odd sequence where i=1 increases.


Gary

[gd_barnes]

I found the lowest odd sequence where i=2 > i=1 > i=0.

So here is what we have:

945 = 3^3*5*7 is the lowest where i=1 > i=0

11025 = 3^2*5^2*7^2 is the lowest where i=2 > i=1 > i=0

Both of these have been verified.

99225 = 3^4*5^2*7^2 is an interesting one. It is the lowest odd sequence that I am personally aware of (not known for sure) that never drops below its starting value. According to the DB, it merges at i=5 with sequence 46758 i=6.

[R. Gerbicz]

Originally Posted by gd_barnes

99225 = 3^4*5^2*7^2 is an interesting one. It is the lowest odd sequence that I am personally aware of (not known for sure) that never drops below its starting value. According to the DB, it merges at i=5 with sequence 46758 i=6.

In fact it is an old conjecture that there is no integer for that the aliquot sequence is strictly increasing.

[Greebley]

BTW, I can't take credit for thinking of the 'very interesting' problem - that was 10metreh in another thread.

Also interesting from that other thread:

There was only one odd number less than 5 million that reached 2^64 without merging - 1334025 and it eventually terminates.

That is how I know the answer is greater than 5 million - I had a simple program that looked for merges below 8 byte numbers (i.e. not using gmp) and factors below 2^26.

As for no strictly increasing sequences, I find the rule of small numbers makes it difficult to surmise. If there are large enough numbers that will average trillions of factors, will they ever decrease? How about googleplex factors? Difficult to know.

[gd_barnes]

Originally Posted by Greebley

As for no strictly increasing sequences, I find the rule of small numbers makes it difficult to surmise. If there are large enough numbers that will average trillions of factors, will they ever decrease? How about googleplex factors? Difficult to know.

Thanks for the extra interesting info.

Here, we may be talking about 2 different things, which may have been your intent. I was only referring to a sequence that never dropped below the sequence value. I think the 2 of you are referring to a sequence where no index ever drops in value.

Although it is an interesting conjecture, how would one ever prove such a thing? As you said, the law of small numbers makes it overly difficult to even debate the point.

An interesting "side project" would be to see how long of a sequence people could come up with that increases consecutivey for the first x number of indexes. It could be divided into both odd and even sequence records.

Correct me if I'm wrong but I am assuming that there has been found an even sequence where all indexes increase up to their currently searched limit. Do you know of one? Do you know the lowest one? It seems like I have seen graphs in the DB where every index appeared to increase but you'd have to look more closely at every index value to be for sure...or you could just look for the factors 2 and 3 in every index. That should do it.


Gary

[kar_bon]

as an example of an only climbing even sequence try 210222:

this seq got at index 282(!!!) a size of 104 digits!
most indices with 2^x * 3^y * 5 (x>=3 and y>=1)

for others search the summary for seqs with small amount of indices and high sizes.

[gd_barnes]

Cool.

Well...it took me 2 indexes at i=284 to eliminate the factor of 5, 3 indexes at i=285 to eliminate 2 powers of 2 and bring it down to 2^2, and 8 indexes at i=290 to bring it down to a single power of 3! That's the lowest power of 2 in 283 indexes since i=2 !!!

Could there be an end in sight? Possibly. The # of small factors is definitely at its lowest point since very early on. There hasn't been a 7, 11, or 13 in quite a while. So there is a chance. If you look at the graph, you can finally start to see a slight flattening. Alas, we have to lose the factor 3.

The next question is: What is the lowest such sequence known?

I think I'll reserve it for a few days and put it on aliqueit.exe to see what it comes up with. If I get a "hard" C97 or higher, I'll probably stop.


Gary

[kar_bon]

just working on 210048:

currently at index 252 with size of 89 ditigs, so quite enough place to push further!

BTW: and smaller than the previous!

PS:the smallest even seq seems to be 19560

[gd_barnes]

I'm now done with 210222. I encounted a hard C106 during the day at i=295. This is an excellent example of the few number of factors that most of the indexes now have. All that was found was 2^3*3. There is likely no other factor < 35 digits. (Several of the indexes had dropped to 2^2*3.)

I got rid of a lot of small factors so if someone has some serious crunching power that they would like to dedicate to it, they might be able to crack this one and make it go down for at least one index.

It would be interesting to get a listing together of all known increase-only sequences. I think people would find them interesting to crack.


Gary