Sequences >1M and < 5M

[pakaran]

Originally Posted by ChristianB

We could do the same as for sequences <1M here: http://www.rechenkraft.net/aliquot/AllSeq.html either extend the current page or set up a new page.

But usually you would check with factordb: http://factordb.com/sequences.php.

My question then is whether there's an easier method than typing 1000000, 1000002, 1000004, and so forth, into the page (especially considering that the set of open sequences isn't particularly dense).

[pakaran]

On another note, is lines coming up "new" on factordb a reliable indication that the sequence has not merged?

I assume that this means that the numbers in the sequence are encountered for the first time, since I got this when I entered my mobile phone number with area code, and also the number 1654866 (which I have my eye on for when I give up on my current one).

Honestly, I found that just via a keyboard smash, so I might have been pessimistic in my last post.

[ChristianB]

I don't know if there already is a list for open sequences >1M and <5M but it is easy to compute one using a script and factordb. Problem here is that factordb limits the amount of queries you can do within an hour. I have a bash script that can be easily adapted to do this and collect the current state in a local sqlite database. This can be used to populate a new AllSeq.html page.

[Batalov]

Originally Posted by pakaran

My question then is whether there's an easier method than typing 1000000, 1000002, 1000004, and so forth, into the page (especially considering that the set of open sequences isn't particularly dense).

The first thing that you should absolutely do is run precisely one iteration using own tools - without bombarding factordb.com with queries. One can do that in Pari:

(09:25) gp > forstep(i=1000000,1000100,2,a=sigma(i)-i;if(a>=i,print(i" "a)))
1000000 1480437
1000002 1000014
1000008 2088792
1000014 1000026
1000020 2201388
1000026 1241136
1000032 1866720
1000038 1154058
1000040 1350040
1000044 1527936
1000048 1214592
1000050 1544430
1000056 1500144
1000060 1169156
1000062 1476594
1000064 1149076
1000068 1333452
1000074 1000086
1000076 1322356
1000080 2452080
1000086 1087338
1000090 1269422
1000092 1333484
1000098 1364238
1000100 1215904

Here, you can see that sequences that decrease in i2 are of course of no interest, as well as a few sequences can be spotted that also need to be excluded (even in this toy example): you remove from the search set (survivors in 1st column) all members of the second column. The you can run survivors with a 10-iteration similar script in Pari, and trim again.

Only after that you can proceed with something like wget "http://factordb.com/sequences.php?se=1&aq=1000008&action=last&fr=0&to=100" and parsing the output. As was already mentioned, the database will limit your "DDoS attack" after a few hundred sequences. It can block you by IP, too.

[pakaran]

Thanks, I hadn't thought of those issues at all. I saw the maximum CPU time per hour, but not that there was a maximum number of queries.

Pari is in my distro, and I've installed it. It might make an interesting programming exercise.

[Happy5214]

If it's helpful at all, here's a list of open sequences between 1,000,000 and 1,005,000 that I compiled while working on a project similar to this one.

[chris2be8]

Originally Posted by Batalov

The first thing that you should absolutely do is run precisely one iteration using own tools - without bombarding factordb.com with queries.

You probably want to start by running one iteration for every number in the range 500,000 to 1,000,000 and keeping a list of all cases where the next step is >= 1,000,000. That should be all the numbers that are part of a smaller sequence, so can be added to the "not interesting" list.

That assumes a sequence can't grow by more that a factor of 2 in one step. If that's wrong start at 1,000,000/x where x is the largest possible ratio a sequence can grow by.

Chris

[ugly2dog]

Originally Posted by pakaran

On another note, is lines coming up "new" on factordb a reliable indication that the sequence has not merged?

Short answer, no.
If no one has queried the sequence since factors have been entered, then they show as new.
Or if Syd rebuilds the index or deletes it [broken sequence].
Many of the sequences above ~1.2M range come up as new lines even though they have been in the factordb for months/years.
The first 100 open sequences above 1M is attached.
I am still actively working getting all sequences up to at least 100 digits. I have 350,000+ lines from over 5000 sequences to upload to the db when I get the time.
I am also working on running the lower end of the 1M range to 120 digits. These I try to add these ever week.

[ugly2dog]

Originally Posted by chris2be8

You probably want to start by running one iteration for every number in the range 500,000 to 1,000,000 and keeping a list of all cases where the next step is >= 1,000,000. That should be all the numbers that are part of a smaller sequence, so can be added to the "not interesting" list.

Initially I started by making a list from 2 to 5,000,000 excluding primes, and ran each sequence until it merged/ended or reach 30 digits.
Marking all the found numbers off the list and saved the 30 digit number to check for possible merges higher up.
Then I checked what I had marked as open with the list <1M.
I then downloaded any sequence <1M that was on my list but not open. I marked all the members of that off my list as well.
I eventually downloaded all open sequences under 1M and marked all those numbers off the list as well.
Some sequences get taken over when a lower sequence drops down into that range (4115300 was past 95 digits and it got clobbered by 213150:I1676). I than ran ever open sequence up to 50, 70, 90 digits.
I checked any that merge\ended to see if they needed to be submitted to the db.
I also check(ed) any sequence that drop below 20 digits to see if it merges in the db.
You probably don't need to go to all the effort, but you may find a sequence that is open that I don't have on my list. Currently 38,318 open sequences from 1000152 to 4999980, but I haven't run a check lately to see if any sequences have ended lately and need to be entered.

[pakaran]

Originally Posted by chris2be8

You probably want to start by running one iteration for every number in the range 500,000 to 1,000,000 and keeping a list of all cases where the next step is >= 1,000,000. That should be all the numbers that are part of a smaller sequence, so can be added to the "not interesting" list.

That assumes a sequence can't grow by more that a factor of 2 in one step. If that's wrong start at 1,000,000/x where x is the largest possible ratio a sequence can grow by.

Chris

It is possible if the number is VERY smooth. I doubt it will happen often for large numbers, but have no intuition on what the maximum growth might be.

[Happy5214]

Originally Posted by pakaran

It is possible if the number is VERY smooth. I doubt it will happen often for large numbers, but have no intuition on what the maximum growth might be.

Growth by at least a ratio of 2 is mandatory for sequences with a driver of 2^3 * 3 * 5. For the simplest case, where the only higher exponent is on the 2, the approximate growth rate can be found with these formulae (x, y, and z are non-2 prime factors, and n is the exponent on the 2):



and so on, with that result plugged into:



So it is much more likely than you think.