20200804, 13:58  #386 
"Ed Hall"
Dec 2009
Adirondack Mtns
3375_{10} Posts 
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). 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). 
20200804, 17:40  #387 
Oct 2011
2^{2}×3×29 Posts 
@Happy5214 : Yes, I had noticed for the presence of 797161 but I hadn't seen that 797161 = (2*3985811) !
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 37digit 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). 
20200804, 18:03  #388 
Oct 2011
2^{2}·3·29 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 ! 
20200804, 18:49  #389  
"Ed Hall"
Dec 2009
Adirondack Mtns
3^{3}·5^{3} Posts 
Quote:
In the meantime, I have decided to color in the transparent cells for base 2310 after all. 

20200805, 02:34  #390  
"Ed Hall"
Dec 2009
Adirondack Mtns
110100101111_{2} Posts 
Quote:
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 ). 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. 

20200805, 07:06  #391 
Oct 2011
2^{2}×3×29 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... 
20200805, 08:40  #392 
Oct 2011
15C_{16} 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 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... 
20200805, 14:22  #393  
"Ed Hall"
Dec 2009
Adirondack Mtns
3^{3}×5^{3} Posts 
Quote:
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 ). Code:
prime 50077 shows up 4 times ( 74:i1145, 107:i1, 214:i1, 214:i2 ). 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 ). 

20200805, 18:44  #394 
Oct 2011
2^{2}·3·29 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 !!! 
20200808, 00:18  #395 
"Ed Hall"
Dec 2009
Adirondack Mtns
D2F_{16} Posts 

20200808, 14:05  #396 
Oct 2011
2^{2}×3×29 Posts 
Thank you very much.
There will be even more colors in the update in a few days ! 
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