Misc. sequences
 

I thought I'd start a thread for misc. sequences since some of us like to "blaze our own trail" for our own entertainment at times.

I've done all of the following aliquot sequences to >= 80 digits:

All primorals to 31#.

All powers of 10 to 10^60.

All digits of PI up to 61 digits.

There was a fair amount of work already in the DB done by others on the first two above, especially on the powers of 10. I started many new ones but extended many others. As far as I can tell, I was the first to do the digits of PI past 7 digits.

The last one was is kind of crazy but I like doing off-the-wall stuff to keep from getting bored. lol To clarify it, I did sequences 3, 31, 314, 3141, 31415, 314159, 3141592, etc. up to 61 digits of PI. All were taken to indexes exceeding 80 digits.

There were some interesting finds on these. The most interesting are two consective very large even sequences of PI that terminated in a prime. They are:

3141592653589793238462643383279502 (34 digits) terminated at i=618 with a prime of 41.
31415926535897932384626433832795028 (35 digits) terminated at i=529 with a prime of 20422951.


The 2nd of the two above technically merged with sequence 14952 towards the end, which has to be the most remarkable sequence I've ever seen. It increases for 13 consecutive indexes before terminating in an 8-digit prime! I did not think that was possible!

The sequences never cease to entertain me in some new way. :smile:

For historical reference, I'd encourage others to post their unusual sequence searches in this thread, mainly the very large ones. Perhaps some strange and new pattern of factors or merges will come out of it. Eventually I'd like to see all powers of 10 filled in to 10^100 up to 100 or 120 digits. The even powers are easy because they quickly become odd and so terminate quickly. It is the odd powers that are more useful.

Here's something for the "odds and ends" thread: What is the LARGEST even sequence that terminates that never becomes odd until towards the very end?

Admins: If you think I'm going too far off the project criteria here, feel free to give me a slap on the hand. :-) I just want to make sure that this historical info. is made public in case others may be interested in it in the future.


Gary

I was going to make a thread just like this a while ago, but never got around to it.
I took 10^15 (which is the lowest power of 10 to not have terminated yet or merged with a seq low enough to be worked by normal efforts, like [URL="http://factordb.com/search.php?se=1&aq=10%5E9"]10^9[/URL]; 10^15 merges to [URL="http://factordb.com/search.php?se=1&aq=823966192835552"]823966192835552[/URL]) to a size of 113 a while ago. It's just got the 2^3 guide right now, but it's big enough to be rather slow at doing anything.

Since the DB does not show sequences >= 100M in its 'sequence overview' list, here are some highest and lowest records related to the above:

Primorals:

Smallest non-terminating sequence: 11# [2310] (merges with 1578)

Largest terminating sequence: 7# [210]

Highest height before terminating: 4 by 7# :smile:

Longest sequences:
i=7258 by 11# (merges with 1578)
i=2122 by 13# (merges with 425052)
i=914 by 23# (longest with no merge < 1M)


Powers of 10 to 10^60:

Smallest non-terminating sequence: 10^9

Largest odd-power terminating sequence: 10^23

Odd-power highest height before terminating: 70 by 10^13 (!)

Longest sequence: i=2129 by 10^35


PI to 61 digits:

Smallest non-terminating sequence: 3141592

Largest even terminating sequence: 31415926535897932384626433832795028 (35 digits)

Even sequence highest height before terminating: 52 by 31415926535897932384626433832795028

Longest sequences:
i=5794 by 3141592 (merges with 8760)
i=2903 by 31415926535897932 (merges with 109480)
i=2590 by 31415926535897932384626 (merges with 77392)
i=2020 by 3141592653589793238 (longest with no merge < 1M)

[quote=Mini-Geek;190591]I was going to make a thread just like this a while ago, but never got around to it.
I took 10^15 (which is the lowest power of 10 to not have terminated yet or merged with a seq low enough to be worked by normal efforts, like [URL="http://factordb.com/search.php?se=1&aq=10%5E9"]10^9[/URL]; 10^15 merges to [URL="http://factordb.com/search.php?se=1&aq=823966192835552"]823966192835552[/URL]) to a size of 113 a while ago. It's just got the 2^3 guide right now, but it's big enough to be rather slow at doing anything.[/quote]

Ah, very good. I knew there was a fair amount of work being done on some of the lower powers of 10. I got into it a lot more above 10^20. 10^23 was my most entertaining one. It's the first sequence I've ever had terminate after reaching at least 70 digits.

[quote=gd_barnes;190588]
Admins: If you think I'm going too far off the project criteria here, feel free to give me a slap on the hand. :-) I just want to make sure that this historical info. is made public in case others may be interested in it in the future.
Gary[/quote]

I find these side excursions interesting (like this and the odd sequences). I think you should post them.

Tim, since you're an admin here now, can you correct something in my 2nd posting here?:

For powers of 10 the following statement:

Smallest non-terminating sequence: 10^15

Should say:

Smallest non-terminating sequence: 10^9


I'm not sure why I thought 10^15 was the smallest non-terminating sequence.

Feel free to delete this post after making the correction.


Thanks,
Gary

I've fixed it, but on closer examination you may want to list both. I'll explain...
[quote=gd_barnes;190984] I'm not sure why I thought 10^15 was the smallest non-terminating sequence.[/quote]
Probably from this:
[quote=Mini-Geek;190591]I took 10^15 (which is the lowest power of 10 to not have terminated yet or merged with a seq low enough to be worked by normal efforts, like [URL="http://factordb.com/search.php?se=1&aq=10%5E9"]10^9[/URL]; 10^15 merges to [URL="http://factordb.com/search.php?se=1&aq=823966192835552"]823966192835552[/URL]) to a size of 113 a while ago. It's just got the 2^3 guide right now, but it's big enough to be rather slow at doing anything.[/quote]
10^9 merges to 31240, and so is covered by more conventional methods of searching (though it's still only at size 101). The next smallest power of 10 to not terminate is 10^15, which merges to a rather large number that would not be searched by normal methods. They're each lowests for their own sort of thing. Perhaps each should have a mention? If you agree with me, let me know what you want it to say and I'll edit it. e.g. Smallest non-terminating sequence with no merges under 1M: 10^15
A similar listing for other sequence categories (x#, pi) may be useful.

Actually, it is how I intended it to be now that you've corrected it. But you make a good point and it is why I listed the top 3-4 longest sequences on a couple of them until I came to the first one with no merge < 1M. I'll look into it and perhaps come up with some additional categories for each of the 3 listings.

There's no limit to the # of possibilities for such types of things. I seem to recall a little while back that I did most of the powers of 3 up to 3^20 but I don't remember how high I searched them.

I've done the same with [URL="http://antwrp.gsfc.nasa.gov/htmltest/gifcity/e.1mil"]e number[/URL]
All digits of "e" up to 100 digits. (2, 27, 271, 2718, 27182, ...). No merges are those with 9, 21, 26, 35, 44, 57, 58, 60, 67, 68, 70, 71, 73, 82 and 91 digits.
All digits of decimal part of e up to 100 digits (7, 71, 718, 7182, 71828, ...). No merges are those with 9, 16, 22, 33, 67, 77, 87, 95 and 97 digits.
Results are in DB.

Excellent. Someone else blazing his own path.

Question: Many of the "e" aliquot sequences are not searched to 100 digits. This includes several "smaller" sequences. How come they are being stopped when they drop below their beginning value? That's the only way that they will terminate.

I feel inspired to run a sequence purely for silliness.

So I will run the Look and Say Sequence up to 80 digits.

It goes:

[code]
1 1
2 11
3 21
4 1211
5 111221
6 312211
7 13112221
8 1113213211
9 31131211131221
10 13211311123113112211
11 11131221133112132113212221
12 3113112221232112111312211312113211
13 1321132132111213122112311311222113111221131221
14 11131221131211131231121113112221121321132132211331222113112211
15 311311222113111231131112132112311321322112111312211312111322212311322113212221
16 132113213221133112132113311211131221121321131211132221123113112221131112311332111213211322211312113211
17 11131221131211132221232112111312212321123113112221121113122113111231133221121321132132211331121321231231121113122113322113111221131221
18 31131122211311123113321112131221123113112211121312211213211321322112311311222113311213212322211211131221131211132221232112111312111213111213211231131122212322211331222113112211
19 1321132132211331121321231231121113112221121321132122311211131122211211131221131211132221121321132132212321121113121112133221123113112221131112311332111213122112311311123112111331121113122112132113213211121332212311322113212221
20 11131221131211132221232112111312111213111213211231132132211211131221131211221321123113213221123113112221131112311332211211131221131211132211121312211231131112311211232221121321132132211331121321231231121113112221121321133112132112312321123113112221121113122113121113123112112322111213211322211312113211
21 311311222113111231133211121312211231131112311211133112111312211213211312111322211231131122211311122122111312211213211312111322211213211321322113311213212322211231131122211311123113223112111311222112132113311213211221121332211211131221131211132221232112111312111213111213211231132132211211131221232112111312211213111213122112132113213221123113112221131112311311121321122112132231121113122113322113111221131221
22 132113213221133112132123123112111311222112132113311213211231232112311311222112111312211311123113322112132113213221133122112231131122211211131221131112311332211211131221131211132221232112111312111213322112132113213221133112132113221321123113213221121113122123211211131221222112112322211231131122211311123113321112131221123113111231121113311211131221121321131211132221123113112211121312211231131122211211133112111311222112111312211312111322211213211321322113311213211331121113122122211211132213211231131122212322211331222113112211
23 111312211312111322212321121113121112131112132112311321322112111312212321121113122112131112131221121321132132211231131122211331121321232221121113122113121113222123112221221321132132211231131122211331121321232221123113112221131112311332111213122112311311123112112322211211131221131211132221232112111312211322111312211213211312111322211231131122111213122112311311221132211221121332211213211321322113311213212312311211131122211213211331121321123123211231131122211211131221131112311332211213211321223112111311222112132113213221123123211231132132211231131122211311123113322112111312211312111322212321121113122123211231131122113221123113221113122112132113213211121332212311322113212221
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
[/code]

That is, 1, one 1 (11), two ones (21), one 2 one 1 (1211), and so on.

I'm reasonably sure nothing interesting will be learnt from this exercise :D