20210309, 03:05  #969 
"Ed Hall"
Dec 2009
Adirondack Mtns
3,719 Posts 

20210309, 17:00  #970 
"Garambois JeanLuc"
Oct 2011
France
2^{3}×73 Posts 
OK, mergers verified.
I will add the base 58 during the next update, the next weekend. Many thanks ! Last fiddled with by garambois on 20210309 at 17:01 
20210312, 19:42  #971 
"Alexander"
Nov 2008
The Alamo City
5×11^{2} Posts 
Another sequence bites the dust thanks to a collaboration of yafu@home and my Kubuntu Focus laptop. 21^98 terminates at 43.

20210312, 20:20  #972 
Oct 2006
Berlin, Germany
264_{16} Posts 
I will not post every terminated sequence here ;)

20210312, 21:10  #973 
"Alexander"
Nov 2008
The Alamo City
5·11^{2} Posts 
I'm sure you would need your own thread for the number of sequences you terminate/merge, assuming opportunistic people like myself don't poach them first. ;) In all seriousness, base 21 was one I had originally reserved (it's my favorite number), so I'm happy to see another termination there.

20210312, 21:15  #974 
"Alexander"
Nov 2008
The Alamo City
5·11^{2} Posts 
By the way, this thread is now almost a thousand posts long. Can someone stick this? It's now a longrunning subproject and being a sticky thread is long overdue IMO.

20210313, 14:08  #975  
"Garambois JeanLuc"
Oct 2011
France
2^{3}×73 Posts 
Quote:
Thanks ! Thanks also for sticking this thread ! 

20210313, 18:24  #976  
"Garambois JeanLuc"
Oct 2011
France
2^{3}×73 Posts 
Quote:
Henryzz, I don't know exactly what you are looking for, but I have done some work on the subject. I have not found a prime number p for which n = (2 * p)^(180 * k) does not have an abundant dividor for any integer k. But I have found good candidates for some p such that the term of index 1 of sequences that start on n is most likely deficient for k = 1 and even some for k = 1, 2, 3 . Why do I say that my finds are only candidates ? Because I only factored n by considering all its prime factors less than 10^4. But in my opinion, it is unlikely that n becomes abundant if we consider the larger prime numbers of its factorization. Here is the result after about 72 hours of running my program : Code:
b = 2 * 3 k 2 b = 2 * 3 k 3 b = 2 * 3 k 4 b = 2 * 300301 k 2 b = 2 * 390391 k 2 b = 2 * 660661 k 2 b = 2 * 798799 k 2 b = 2 * 930931 k 2 b = 2 * 1179179 k 2 b = 2 * 1360591 k 2 b = 2 * 1623931 k 2 b = 2 * 1939939 k 2 b = 2 * 2372371 k 2 b = 2 * 2402401 k 2 b = 2 * 2482481 k 2 b = 2 * 2548547 k 2 b = 2 * 2662661 k 2 b = 2 * 3233231 k 2 b = 2 * 3333331 k 2 b = 2 * 3663661 k 2 b = 2 * 3723721 k 2 b = 2 * 3873871 k 2 b = 2 * 3993991 k 2 b = 2 * 3993991 k 3 b = 2 * 3993991 k 4 b = 2 * 4792789 k 2 b = 2 * 6066061 k 2 b = 2 * 6276271 k 2 b = 2 * 6516511 k 2 b = 2 * 6846841 k 2 b = 2 * 6846841 k 3 b = 2 * 7087081 k 2 b = 2 * 7417411 k 2 b = 2 * 8368361 k 2 b = 2 * 8558551 k 2 b = 2 * 8558551 k 3 b = 2 * 8558551 k 4 b = 2 * 8708701 k 2 b = 2 * 8786779 k 2 b = 2 * 8978971 k 2 b = 2 * 8998991 k 2 b = 2 * 10420411 k 2 b = 2 * 10498489 k 2 b = 2 * 10840831 k 2 b = 2 * 11111101 k 2 b = 2 * 11221211 k 2 b = 2 * 11297287 k 2 b = 2 * 11791781 k 2 b = 2 * 11791781 k 3 b = 2 * 11791781 k 4 b = 2 * 12192181 k 2 b = 2 * 12222211 k 2 b = 2 * 12438427 k 2 b = 2 * 12552541 k 2 b = 2 * 12642631 k 2 For example, for the first three lines in bold : Code:
b = 2 * 3 k 2 b = 2 * 3 k 3 b = 2 * 3 k 4 And s(n) = s(b^(180 * k)) = s(6^(180 * k)) is abundant only from k = 4. I did not then test for k> 4. The same goes for all the other bases noted in bold : b = 2 * 3993991 (abundant only from k = 4 in this case) b = 2 * 6846841 (abundant only from k = 3 in this case) b = 2 * 8558551 (abundant only from k = 4 in this case) b = 2 * 11791781 (abundant only from k = 4 in this case) And for all the other bases which are not noted in bold, n is deficient only for k = 1 and becomes abundant for k = 2. I carried out this test for the first 853683 prime numbers, that is to say for all the prime numbers p < 13073363. And for all the prime numbers p < 13073363 which do not appear in the table above, we therefore have s(n) = s((2 * p)^(180 * k)) which is abundant (proved by calculation) already from k = 1. So I did not test the k > 1 for all these prime numbers which are not in the table above. An extremely curious finding : It would seem that some prime numbers in the table above are arithmetic progressions. For example, if we look at the prime numbers that appear at the beginning of the table, we can see that : 390391 + 270270 = 660661 And after that : 660661 + 270270 = 930931 Indeed, the prime numbers 390391, 660661 and 930931 appear in the table. But unfortunately, the next one which is also prime does not appear in the table (930931 + 270270 = 1201201 is prime). I do not have time to occupy myself with this question of the presence of arithmetic progressions in this table of prime numbers, but perhaps there are completely new conjectures to formulate ? Henryzz, I don't know if this is what you were looking for ? Do you need me to let the program run any further ? The program I wrote is powerful at finding sequences starting with numbers whose base and exponent have the same parity and which have an abundant index 1 term. We do not need to decompose the term. A week ago, in post #952, I presented the first even exponent i = 2^3 * 3^2 * 5 * 7 such that sequences starting with the number n = 6^(i * k) have a index term 1 abundant for all k and with the following abundant factor d which was suitable for all these sequences : d = 5^2 * 7^2 * 11 * 13 * 19 * 29 * 31 * 37 * 41 * 43 * 61 * 71 * 73 * 127 * 181 * 211 * 281 * 337 * 421 * 631 This gave the conjecture (138) that several of you have subsequently proved. With this new program, in just 10 seconds, I have already found a dozen of exponents that were suitable for base 6. And after 1 hour and 15 minutes of operation, I found all of the following exponents that were suitable : Code:
[720, 1080, 1260, 1440, 1680, 1800, 1980, 2160, 2340, 2520, 2640, 2700, 2772, 2880, 3120, 3240, 3360, 3600, 3780, 3960, 4032, 4200, 4320, 4680, 5040, 5280, 5400, 5760, 5940, 6240, 6480, 6720, 7020, 7200, 7560, 7920, 8100, 8400, 8640, 9360, 9720, 10080, 10800, 11340, 11880, 12600] Code:
b = 6 k=1 exposant e = 720 = 2^4 * 3^2 * 5 i = 720 = 2^4 * 3^2 * 5 b = 6 k=2 exposant e = 1440 = 2^5 * 3^2 * 5 i = 720 = 2^4 * 3^2 * 5 b = 6 k=3 exposant e = 2160 = 2^4 * 3^3 * 5 i = 720 = 2^4 * 3^2 * 5 Sequence multiples pour base b = 6 i = 720 = 2^4 * 3^2 * 5 d = 4132565474853920632953425 = 5^2 * 7 * 11 * 13 * 17 * 19 * 31 * 37 * 41 * 61 * 73 * 97 * 181 * 241 * 577 b = 6 k=1 exposant e = 1080 = 2^3 * 3^3 * 5 i = 1080 = 2^3 * 3^3 * 5 b = 6 k=3 exposant e = 3240 = 2^3 * 3^4 * 5 i = 1080 = 2^3 * 3^3 * 5 Sequence multiples pour base b = 6 i = 1080 = 2^3 * 3^3 * 5 d = 64946984492713940892505843187575 = 5^2 * 7 * 11 * 13 * 19 * 31 * 37 * 41 * 61 * 73 * 109 * 181 * 241 * 271 * 433 * 541 * 2161 b = 6 k=1 exposant e = 1260 = 2^2 * 3^2 * 5 * 7 i = 1260 = 2^2 * 3^2 * 5 * 7 b = 6 k=2 exposant e = 2520 = 2^3 * 3^2 * 5 * 7 i = 1260 = 2^2 * 3^2 * 5 * 7 b = 6 k=3 exposant e = 3780 = 2^2 * 3^3 * 5 * 7 i = 1260 = 2^2 * 3^2 * 5 * 7 Sequence multiples pour base b = 6 i = 1260 = 2^2 * 3^2 * 5 * 7 d = 4888951109215725280985556125250425 = 5^2 * 7^2 * 11 * 13 * 19 * 29 * 31 * 37 * 43 * 61 * 71 * 73 * 127 * 181 * 211 * 421 * 631 * 2521 b = 6 k=2 exposant e = 2880 = 2^6 * 3^2 * 5 i = 1440 = 2^5 * 3^2 * 5 b = 6 k=3 exposant e = 4320 = 2^5 * 3^3 * 5 i = 1440 = 2^5 * 3^2 * 5 Sequence multiples pour base b = 6 i = 1440 = 2^5 * 3^2 * 5 d = 797585136646806682160011025 = 5^2 * 7 * 11 * 13 * 17 * 19 * 31 * 37 * 41 * 61 * 73 * 97 * 181 * 193 * 241 * 577 b = 6 k=1 exposant e = 1680 = 2^4 * 3 * 5 * 7 i = 1680 = 2^4 * 3 * 5 * 7 b = 6 k=2 exposant e = 3360 = 2^5 * 3 * 5 * 7 i = 1680 = 2^4 * 3 * 5 * 7 b = 6 k=3 exposant e = 5040 = 2^4 * 3^2 * 5 * 7 i = 1680 = 2^4 * 3 * 5 * 7 Sequence multiples pour base b = 6 i = 1680 = 2^4 * 3 * 5 * 7 d = 1027478644809677744369628450729130985075 = 5^2 * 7^2 * 11 * 13 * 17 * 29 * 31 * 41 * 43 * 61 * 71 * 97 * 113 * 211 * 241 * 281 * 337 * 421 * 673 * 3361 b = 6 k=1 exposant e = 1800 = 2^3 * 3^2 * 5^2 i = 1800 = 2^3 * 3^2 * 5^2 b = 6 k=2 exposant e = 3600 = 2^4 * 3^2 * 5^2 i = 1800 = 2^3 * 3^2 * 5^2 b = 6 k=3 exposant e = 5400 = 2^3 * 3^3 * 5^2 i = 1800 = 2^3 * 3^2 * 5^2 Sequence multiples pour base b = 6 i = 1800 = 2^3 * 3^2 * 5^2 d = 430549205282764555621103122027375 = 5^3 * 7 * 11 * 13 * 19 * 31 * 37 * 41 * 61 * 73 * 101 * 151 * 181 * 241 * 601 * 1201 * 1801 b = 6 k=1 exposant e = 1980 = 2^2 * 3^2 * 5 * 11 i = 1980 = 2^2 * 3^2 * 5 * 11 b = 6 k=2 exposant e = 3960 = 2^3 * 3^2 * 5 * 11 i = 1980 = 2^2 * 3^2 * 5 * 11 b = 6 k=3 exposant e = 5940 = 2^2 * 3^3 * 5 * 11 i = 1980 = 2^2 * 3^2 * 5 * 11 Sequence multiples pour base b = 6 i = 1980 = 2^2 * 3^2 * 5 * 11 d = 500921715483185619108910812206761117175 = 5^2 * 7 * 11^2 * 13 * 19 * 23 * 31 * 37 * 61 * 67 * 73 * 181 * 199 * 331 * 397 * 661 * 991 * 1321 * 2971 b = 6 k=3 exposant e = 6480 = 2^4 * 3^4 * 5 i = 2160 = 2^4 * 3^3 * 5 Sequence multiples pour base b = 6 i = 2160 = 2^4 * 3^3 * 5 d = 1122456169041960873091755904130496194140035570711503206523075 = 5^2 * 7 * 11 * 13 * 17 * 19 * 31 * 37 * 41 * 61 * 73 * 97 * 109 * 163 * 181 * 241 * 271 * 433 * 541 * 577 * 811 * 1297 * 1621 * 2161 * 2593 * 3889 * 6481 b = 6 k=1 exposant e = 2340 = 2^2 * 3^2 * 5 * 13 i = 2340 = 2^2 * 3^2 * 5 * 13 b = 6 k=2 exposant e = 4680 = 2^3 * 3^2 * 5 * 13 i = 2340 = 2^2 * 3^2 * 5 * 13 b = 6 k=3 exposant e = 7020 = 2^2 * 3^3 * 5 * 13 i = 2340 = 2^2 * 3^2 * 5 * 13 Sequence multiples pour base b = 6 i = 2340 = 2^2 * 3^2 * 5 * 13 d = 2592519558190028920204970920911756521975 = 5^2 * 7 * 11 * 13^2 * 19 * 31 * 37 * 53 * 61 * 73 * 79 * 131 * 157 * 181 * 313 * 937 * 1171 * 2341 * 6553 b = 6 k=3 exposant e = 7560 = 2^3 * 3^3 * 5 * 7 i = 2520 = 2^3 * 3^2 * 5 * 7 Sequence multiples pour base b = 6 i = 2520 = 2^3 * 3^2 * 5 * 7 d = 149724897769239702501470160794374161071134869124345288345668279421025 = 5^2 * 7^2 * 11 * 13 * 19 * 29 * 31 * 37 * 41 * 43 * 61 * 71 * 73 * 109 * 127 * 181 * 211 * 241 * 271 * 281 * 337 * 379 * 421 * 433 * 541 * 631 * 757 * 1009 * 2161 * 2521 * 7561 b = 6 k=1 exposant e = 2640 = 2^4 * 3 * 5 * 11 i = 2640 = 2^4 * 3 * 5 * 11 b = 6 k=2 exposant e = 5280 = 2^5 * 3 * 5 * 11 i = 2640 = 2^4 * 3 * 5 * 11 b = 6 k=3 exposant e = 7920 = 2^4 * 3^2 * 5 * 11 i = 2640 = 2^4 * 3 * 5 * 11 Sequence multiples pour base b = 6 i = 2640 = 2^4 * 3 * 5 * 11 d = 1564216444318060267573094415936051275 = 5^2 * 7 * 11^2 * 13 * 17 * 23 * 31 * 41 * 61 * 67 * 89 * 97 * 241 * 331 * 661 * 881 * 1321 * 5281 b = 6 k=1 exposant e = 2700 = 2^2 * 3^3 * 5^2 i = 2700 = 2^2 * 3^3 * 5^2 b = 6 k=3 exposant e = 8100 = 2^2 * 3^4 * 5^2 i = 2700 = 2^2 * 3^3 * 5^2 Sequence multiples pour base b = 6 i = 2700 = 2^2 * 3^3 * 5^2 d = 696329743891563806780087534740162625 = 5^3 * 7 * 11 * 13 * 19 * 31 * 37 * 61 * 73 * 101 * 109 * 151 * 181 * 271 * 541 * 601 * 1201 * 1801 b = 6 k=1 exposant e = 2772 = 2^2 * 3^2 * 7 * 11 i = 2772 = 2^2 * 3^2 * 7 * 11 b = 6 k=2 exposant e = 5760 = 2^7 * 3^2 * 5 i = 2880 = 2^6 * 3^2 * 5 b = 6 k=3 exposant e = 8640 = 2^6 * 3^3 * 5 i = 2880 = 2^6 * 3^2 * 5 Sequence multiples pour base b = 6 i = 2880 = 2^6 * 3^2 * 5 d = 919615662553768104530492711825 = 5^2 * 7 * 11 * 13 * 17 * 19 * 31 * 37 * 41 * 61 * 73 * 97 * 181 * 193 * 241 * 577 * 1153 b = 6 k=1 exposant e = 3120 = 2^4 * 3 * 5 * 13 i = 3120 = 2^4 * 3 * 5 * 13 b = 6 k=2 exposant e = 6240 = 2^5 * 3 * 5 * 13 i = 3120 = 2^4 * 3 * 5 * 13 b = 6 k=3 exposant e = 9360 = 2^4 * 3^2 * 5 * 13 i = 3120 = 2^4 * 3 * 5 * 13 Sequence multiples pour base b = 6 i = 3120 = 2^4 * 3 * 5 * 13 d = 1284529589404309867171183810386986553775 = 5^2 * 7 * 11 * 13^2 * 17 * 31 * 41 * 53 * 61 * 79 * 97 * 131 * 157 * 241 * 313 * 521 * 1249 * 2341 * 3121 b = 6 k=3 exposant e = 9720 = 2^3 * 3^5 * 5 i = 3240 = 2^3 * 3^4 * 5 Sequence multiples pour base b = 6 i = 3240 = 2^3 * 3^4 * 5 d = 27148037961226222991113680636686083867943088958727935867644872425 = 5^2 * 7 * 11 * 13 * 19 * 31 * 37 * 41 * 61 * 73 * 109 * 163 * 181 * 241 * 271 * 433 * 487 * 541 * 811 * 1297 * 1621 * 2161 * 2593 * 3889 * 4861 * 6481 * 9721 b = 6 k=2 exposant e = 6720 = 2^6 * 3 * 5 * 7 i = 3360 = 2^5 * 3 * 5 * 7 b = 6 k=3 exposant e = 10080 = 2^5 * 3^2 * 5 * 7 i = 3360 = 2^5 * 3 * 5 * 7 Sequence multiples pour base b = 6 i = 3360 = 2^5 * 3 * 5 * 7 d = 198303378448267804663338290990722280119475 = 5^2 * 7^2 * 11 * 13 * 17 * 29 * 31 * 41 * 43 * 61 * 71 * 97 * 113 * 193 * 211 * 241 * 281 * 337 * 421 * 673 * 3361 b = 6 k=2 exposant e = 7200 = 2^5 * 3^2 * 5^2 i = 3600 = 2^4 * 3^2 * 5^2 b = 6 k=3 exposant e = 10800 = 2^4 * 3^3 * 5^2 i = 3600 = 2^4 * 3^2 * 5^2 Sequence multiples pour base b = 6 i = 3600 = 2^4 * 3^2 * 5^2 d = 788670033040908691634515988885664479014123375 = 5^3 * 7 * 11 * 13 * 17 * 19 * 31 * 37 * 41 * 61 * 73 * 97 * 101 * 151 * 181 * 241 * 401 * 577 * 601 * 1201 * 1801 * 4801 b = 6 k=3 exposant e = 11340 = 2^2 * 3^4 * 5 * 7 i = 3780 = 2^2 * 3^3 * 5 * 7 Sequence multiples pour base b = 6 i = 3780 = 2^2 * 3^3 * 5 * 7 d = 82404142607783838376929183648309830176928742845471755163471825 = 5^2 * 7^2 * 11 * 13 * 19 * 29 * 31 * 37 * 43 * 61 * 71 * 73 * 109 * 127 * 163 * 181 * 211 * 271 * 379 * 421 * 541 * 631 * 757 * 811 * 1621 * 2269 * 2521 * 7561 b = 6 k=3 exposant e = 11880 = 2^3 * 3^3 * 5 * 11 i = 3960 = 2^3 * 3^2 * 5 * 11 Sequence multiples pour base b = 6 i = 3960 = 2^3 * 3^2 * 5 * 11 d = 15657616943680432898365837997947429516858002526388553596160425 = 5^2 * 7 * 11^2 * 13 * 19 * 23 * 31 * 37 * 41 * 61 * 67 * 73 * 89 * 109 * 181 * 199 * 241 * 271 * 331 * 397 * 433 * 541 * 661 * 991 * 1321 * 2161 * 2377 * 2971 b = 6 k=1 exposant e = 4032 = 2^6 * 3^2 * 7 i = 4032 = 2^6 * 3^2 * 7 b = 6 k=1 exposant e = 4200 = 2^3 * 3 * 5^2 * 7 i = 4200 = 2^3 * 3 * 5^2 * 7 b = 6 k=2 exposant e = 8400 = 2^4 * 3 * 5^2 * 7 i = 4200 = 2^3 * 3 * 5^2 * 7 b = 6 k=3 exposant e = 12600 = 2^3 * 3^2 * 5^2 * 7 i = 4200 = 2^3 * 3 * 5^2 * 7 Sequence multiples pour base b = 6 i = 4200 = 2^3 * 3 * 5^2 * 7 d = 2616729487213159500641282093659007108501832335890375 = 5^3 * 7^2 * 11 * 13 * 29 * 31 * 41 * 43 * 61 * 71 * 101 * 151 * 211 * 241 * 281 * 337 * 421 * 601 * 701 * 1051 * 1201 * 4201 * 6301 Code:
Sequence multiples pour base b = 6 i = 720 = 2^4 * 3^2 * 5 d = 4132565474853920632953425 = 5^2 * 7 * 11 * 13 * 17 * 19 * 31 * 37 * 41 * 61 * 73 * 97 * 181 * 241 * 577 Base 6 sequences starting with 6^(720 * k) are increasing at least from index 1 to 2. And we can see that the GCD (with only primes < 10^4) of all the terms of index 1 of sequences 6^(720 * k) for k from 1 to 3 is d = 4132565474853920632953425 = 5^2 * 7 * 11 * 13 * 17 * 19 * 31 * 37 * 41 * 61 * 73 * 97 * 181 * 241 * 577. Thus, the same program found conjecture (138) by displaying the following line : Code:
Sequence multiples pour base b = 6 i = 2520 = 2^3 * 3^2 * 5 * 7 d = 149724897769239702501470160794374161071134869124345288345668279421025 = 5^2 * 7^2 * 11 * 13 * 19 * 29 * 31 * 37 * 41 * 43 * 61 * 71 * 73 * 109 * 127 * 181 * 211 * 241 * 271 * 281 * 337 * 379 * 421 * 433 * 541 * 631 * 757 * 1009 * 2161 * 2521 * 7561 The same phenomenon occurs if I run the program with bases 12 and 24 for which we did not have a suitable exponent in post #921. After a few minutes, we can formulate a multitude of conjectures similar to conjecture (138) also for these bases. My new holy grail : On the other hand : nothing for odd bases which are prime numbers, like the base 3 for example. Right now, I'm letting the program run. This is my new holy grail : a base 3 sequence with an odd exponent whose term at index 1 would be abundant ! 

20210313, 21:12  #977 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2·5·587 Posts 
I suspect if i is a suitable multiplier for a base to make it abundant then any multiple of i will be.
The thing I was wanting was an example such as 300301 that is deficient for (2 * p)^(180 * k). You have found many of them although they aren't that common. I suspect that for any multiplier there will be counter examples although I would imagine there will be a multiplier suitable for any p. 
20210314, 16:23  #978  
"Garambois JeanLuc"
Oct 2011
France
1110_{8} Posts 
Quote:
I tried my program with the base 30. Nothing at all after 24 hours of calculation !!! While it only took a few seconds for bases 6, 12 and 24 ... I will be doing some other tests in the next few days and try to figure out which are the even and odd bases for which it is very difficult to find sequences (with base and exponent of the same parity) that have abundant index 1 terms ! 

20210314, 16:26  #979 
"Garambois JeanLuc"
Oct 2011
France
2^{3}·73 Posts 

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