Aliquot sequences that start on the integer powers n^i - Page 36

EdH · 2020-08-04, 13:58

Here is a full set of all the bases represented on the page. I forgot to add the latest lines of the sequences that have been updated within the last couple of days. I will try to fix that soon.

Here are the two primes mentioned above as listed in the new files:

base3primes 398581:

Code:

prime 398581 shows up 18 times (26:i1, 26:i2, 52:i1, 52:i2, 78:i1, 78:i2, 104:i1, 104:i2, 130:i1, 130:i2, 156:i1, 156:i2, 182:i1, 182:i2, 208:i1, 208:i2, 234:i1, 234:i2).

base3primes 797161:

Code:

prime 797161 shows up 19 times (13:i1, 26:i1, 39:i1, 52:i1, 65:i1, 78:i1, 91:i1, 104:i1, 117:i1, 130:i1, 143:i1, 156:i1, 169:i1, 182:i1, 195:i1, 208:i1, 221:i1, 234:i1, 247:i1).

garambois · 2020-08-04, 17:40

@Happy5214 : Yes, I had noticed for the presence of 797161 but I hadn't seen that 797161 = (2*398581-1) !
I also just noticed that 3^13 ends directly on the prime number 797161 at index 1.
Thank you for checking the conjecture even further !

@EdH : Thank you for these precious tables !
I'm going to look at them very closely tomorrow.

But in just a few minutes I have already noticed exactly the same phenomenon with the 37-digit prime number : 1535090713229126909942383374434289901 which is in the decomposition of all terms in index 1 and 2 of all the sequences that start with 3^(206*k), k integer.
And exactly in the same way, 3^103 also ends directly on a prime number (of 49 digits) : 69575965298821529689922251835887181478451547013. On the other hand, I haven't yet found the relation between these two prime numbers, as Happy5214 did for the previous case !
It's this line of the table that made me see this new similar case :

Code:

prime 1535090713229126909942383374434289901 shows up 2 times (206:i1, 206:i2).

I just checked this new conjecture up to 3^824 and it works fine !

garambois · 2020-08-04, 18:03

@EdH : I think it would be extremely efficient to generate the tables for the different bases by making only the prime numbers >= 10^4 appear and which in addition to that, also appear at several indexes in the same sequence !
I'll be able to do this myself in a few days, but maybe for you it's not too complicated and I'll see the results a few days in advance...
No problem for me if you want to stop now and not generate these new tables, because all this is really a lot of work and requires really a lot of time !

EdH · 2020-08-04, 18:49

Quote:

Originally Posted by garambois

@EdH : I think it would be extremely efficient to generate the tables for the different bases by making only the prime numbers >= 10^4 appear and which in addition to that, also appear at several indexes in the same sequence !
I'll be able to do this myself in a few days, but maybe for you it's not too complicated and I'll see the results a few days in advance...
No problem for me if you want to stop now and not generate these new tables, because all this is really a lot of work and requires really a lot of time !

I can very easily run everything again for all primes > 10^4, but as to limiting the lists to only those that show a repetition within a sequence, it might take a bit more work. Let me think about this a bit.

In the meantime, I have decided to color in the transparent cells for base 2310 after all.

EdH · 2020-08-05, 02:34

Quote:

Originally Posted by garambois

@EdH : I think it would be extremely efficient to generate the tables for the different bases by making only the prime numbers >= 10^4 appear and which in addition to that, also appear at several indexes in the same sequence !
I'll be able to do this myself in a few days, but maybe for you it's not too complicated and I'll see the results a few days in advance...
No problem for me if you want to stop now and not generate these new tables, because all this is really a lot of work and requires really a lot of time !

I think I have it working, but we'll have to see. It definitely knocks the size of the files down. Here is the entire base2primes for the new filtering (if I have it right: primes >10^4 that repeat within a sequence):

Code:

prime 10111 shows up 4 times ( 76:i10, 416:i61, 416:i81, 506:i6 ).
prime 10613 shows up 3 times ( 148:i3, 148:i23, 516:i104 ).
prime 10667 shows up 3 times ( 111:i19, 405:i47, 405:i59 ).
prime 15121 shows up 5 times ( 220:i15, 270:i1, 309:i27, 540:i1, 540:i2 ).
prime 37517 shows up 3 times ( 219:i28, 219:i56, 467:i8 ).

Although base 2 shows only five primes meeting the criteria, base 3 has a few more at 1417.

I'm running a full set for all the tables in hopes it will be done when I get up in the morning. When finished, I'll upload the files so you can see what you think.

garambois · 2020-08-05, 07:06

@EdH :
A lot of thanks for the base 2310.
A lot of thanks for your new effort ! I look forward to the results !
For my part, I'm trying to reproduce your calculations so that the program execution time is reasonable...

garambois · 2020-08-05, 08:40

I think I am now finally able to formulate a general conjecture that encompasses the two little conjectures stated in posts #384 and #387.

General conjecture :

s(n) = sigma(n)-n
If p = s(3^i) is a prime, then we have :
- s(3^(2i)) = m * p and s(s(3^(2i))) = m * r, where r is any integer.
- s(3^(2i * k)) = m * p * u and s(s(3^(2i * k))) = m * t, with u and t any integers but p and m which remain the same whatever k integer k>=1 for a given i.

I hope I have made this conjecture clear in English !
For a better understanding, here are some numerical examples below (I looked up all the i<=500 such that p = s(3^i) is a prime number) :

Code:

i = 3    p = 13                                                 m = 28 = 2^2 * 7
i = 7    p = 1093                                               m = 2188 = 2^2 * 547
i = 13   p = 797161                                             m = 1594324 = 2^2 * 398581
i = 71   p = 3754733257489862401973357979128773                 m = 7509466514979724803946715958257548 = 2^2 * 853 * 2131 * 82219 * 3099719989 * 4052490063499
i = 103  p = 6957596529882152968992225251835887181478451547013  m = 13915193059764305937984450503671774362956903094028 = 2^2 * 619 * 3661040653 * 1535090713229126909942383374434289901

If we take for example i=71, we have :
i=71 p=3754733257489862401973357979128773 m=7509466514979724803946715958257548=2^2*853*2131*82219*309919989*4052490063499
So, we can say that for all sequences that begin with 3^(2*71 * k) = 3^(142k), with k being an integer, we will find the factor p*m in the decomposition of the term at index 1 and we will find the factor m in the decomposition of the term at index 2.

I'm not quite sure how to try to demonstrate this conjecture yet, I haven't spent any time on it. Either she's already known. Otherwise, it shouldn't be very difficult to prove it for someone who's used to this kind of problem...

EdH · 2020-08-05, 14:22

Quote:

Originally Posted by garambois

@EdH :
A lot of thanks for the base 2310.
A lot of thanks for your new effort ! I look forward to the results !
For my part, I'm trying to reproduce your calculations so that the program execution time is reasonable...

Here is a set of (hopefully) all the primes >10^4 that repeat within a single sequence for all the current bases listed on the page. If the prime shows up in other sequences, those are listed as well, whether there are repetitions in the subsequent sequence(s) or not.

Here is a familiar sample from the end of base 3:

Code:

prime 3099719989 shows up 2 times ( 142:i1, 142:i2 ).
prime 3661040653 shows up 2 times ( 206:i1, 206:i2 ).
prime 4052490063499 shows up 2 times ( 142:i1, 142:i2 ).
prime 1535090713229126909942383374434289901 shows up 2 times ( 206:i1, 206:i2 ).

Also in base 3:

Code:

prime 50077 shows up 4 times ( 74:i1145, 107:i1, 214:i1, 214:i2 ).

Here is a full list of the primes I found that show up in Indices 1 and 2 within a sequence:

Code:

base2primes:prime 15121 shows up 5 times ( 220:i15, 270:i1, 309:i27, 540:i1, 540:i2 ).
base3primes:prime 17761 shows up 10 times ( 14:i1550, 28:i2055, 68:i1359, 110:i2051, 140:i1820, 185:i1, 185:i2, 204:i1259, 210:i18, 235:i56 ).
base3primes:prime 50077 shows up 4 times ( 74:i1145, 107:i1, 214:i1, 214:i2 ).
base3primes:prime 82219 shows up 5 times ( 78:i861, 142:i1, 142:i2, 150:i633, 204:i10450 ).
base3primes:prime 398581 shows up 18 times ( 26:i1, 26:i2, 52:i1, 52:i2, 78:i1, 78:i2, 104:i1, 104:i2, 130:i1, 130:i2, 156:i1, 156:i2, 182:i1, 182:i2, 208:i1, 208:i2, 234:i1, 234:i2 ).
base3primes:prime 3099719989 shows up 2 times ( 142:i1, 142:i2 ).
base3primes:prime 3661040653 shows up 2 times ( 206:i1, 206:i2 ).
base3primes:prime 4052490063499 shows up 2 times ( 142:i1, 142:i2 ).
base3primes:prime 1535090713229126909942383374434289901 shows up 2 times ( 206:i1, 206:i2 ).

garambois · 2020-08-05, 18:44

Thank you very much Ed for these precious new tables. I'm going to take a close look at them...

Right now, I have no idea what's going on with the prime number 50077. Either this number escapes conjecture, or the conjecture is still incomplete !!!

EdH · 2020-08-08, 00:18

Quote:

Originally Posted by EdH

. . .
In the meantime, I have decided to color in the transparent cells for base 2310 after all.

They should all be colored in now. All the cofactors that remain have been ECMed to Aliqueit's fullest desires.

garambois · 2020-08-08, 14:05

Thank you very much.
There will be even more colors in the update in a few days !

2020-08-04, 17:40	#387
garambois "Garambois Jean-Luc" Oct 2011 France 649₁₀ Posts	@Happy5214 : Yes, I had noticed for the presence of 797161 but I hadn't seen that 797161 = (2398581-1) ! I also just noticed that 3^13 ends directly on the prime number 797161 at index 1. Thank you for checking the conjecture even further ! @EdH : Thank you for these precious tables ! I'm going to look at them very closely tomorrow. But in just a few minutes I have already noticed exactly the same phenomenon with the 37-digit prime number : 1535090713229126909942383374434289901 which is in the decomposition of all terms in index 1 and 2 of all the sequences that start with 3^(206k), k integer. And exactly in the same way, 3^103 also ends directly on a prime number (of 49 digits) : 69575965298821529689922251835887181478451547013. On the other hand, I haven't yet found the relation between these two prime numbers, as Happy5214 did for the previous case ! It's this line of the table that made me see this new similar case : Code: prime 1535090713229126909942383374434289901 shows up 2 times (206:i1, 206:i2). I just checked this new conjecture up to 3^824 and it works fine !

2020-08-05, 08:40	#392
garambois "Garambois Jean-Luc" Oct 2011 France 1010001001₂ Posts	I think I am now finally able to formulate a general conjecture that encompasses the two little conjectures stated in posts #384 and #387. General conjecture : s(n) = sigma(n)-n If p = s(3^i) is a prime, then we have : - s(3^(2i)) = m * p and s(s(3^(2i))) = m * r, where r is any integer. *- s(3^(2i k)) = m * p * u and s(s(3^(2i * k))) = m * t, with u and t any integers but p and m which remain the same whatever k integer k>=1 for a given i.** I hope I have made this conjecture clear in English ! For a better understanding, here are some numerical examples below (I looked up all the i<=500 such that p = s(3^i) is a prime number) : Code: i = 3 p = 13 m = 28 = 2^2 * 7 i = 7 p = 1093 m = 2188 = 2^2 * 547 i = 13 p = 797161 m = 1594324 = 2^2 * 398581 i = 71 p = 3754733257489862401973357979128773 m = 7509466514979724803946715958257548 = 2^2 * 853 * 2131 * 82219 * 3099719989 * 4052490063499 i = 103 p = 6957596529882152968992225251835887181478451547013 m = 13915193059764305937984450503671774362956903094028 = 2^2 * 619 * 3661040653 * 1535090713229126909942383374434289901 If we take for example i=71, we have : i=71 p=3754733257489862401973357979128773 m=7509466514979724803946715958257548=2^28532131822193099199894052490063499 So, we can say that for all sequences that begin with 3^(271 * k) = 3^(142k), with k being an integer, we will find the factor p*m in the decomposition of the term at index 1 and we will find the factor m in the decomposition of the term at index 2. I'm not quite sure how to try to demonstrate this conjecture yet, I haven't spent any time on it. Either she's already known. Otherwise, it shouldn't be very difficult to prove it for someone who's used to this kind of problem...

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2020-08-04, 18:03	#388
garambois "Garambois Jean-Luc" Oct 2011 France 11·59 Posts	@EdH : I think it would be extremely efficient to generate the tables for the different bases by making only the prime numbers >= 10^4 appear and which in addition to that, also appear at several indexes in the same sequence ! I'll be able to do this myself in a few days, but maybe for you it's not too complicated and I'll see the results a few days in advance... No problem for me if you want to stop now and not generate these new tables, because all this is really a lot of work and requires really a lot of time !

2020-08-05, 07:06	#391
garambois "Garambois Jean-Luc" Oct 2011 France 11·59 Posts	@EdH : A lot of thanks for the base 2310. A lot of thanks for your new effort ! I look forward to the results ! For my part, I'm trying to reproduce your calculations so that the program execution time is reasonable...

2020-08-05, 18:44	#394
garambois "Garambois Jean-Luc" Oct 2011 France 11·59 Posts	Thank you very much Ed for these precious new tables. I'm going to take a close look at them... Right now, I have no idea what's going on with the prime number 50077. Either this number escapes conjecture, or the conjecture is still incomplete !!!

2020-08-08, 14:05	#396
garambois "Garambois Jean-Luc" Oct 2011 France 11×59 Posts	Thank you very much. There will be even more colors in the update in a few days !