Aliquot sequences that start on the integer powers n^i

[EdH]

Well, I am exploring some ideas, but I'll have to see if any work out. One thing going toward identification of these is that, although the term will be different for each sequence, values (comp = factor1 * factor2 * factor3) will all be identical.

[RichD]

Originally Posted by garambois

I also noticed yesterday that the prime 181 was missing from the list of all bases < 200 that are primes. I don't know why we had forgotten 181 ?
I will initialize it.

Oops, I seem to have missed this. I did some work on base 181. I will stop now.

[garambois]

Originally Posted by RichD

Oops, I seem to have missed this. I did some work on base 181. I will stop now.

OK, now I understand better why the work for some exponents was already done !
Thanks.

[EdH]

I have not solved the entire problem with merged duplication entirely, but I have come up with some results. I ran some work on all the primes from all the sequences from base 2 through base 200, that have the first digit of 1, against the entire set. In my attempt to cull those that appeared in merged sequences, I did find a couple triplets where a prime exists within merged sequences and outside of them. I suspect many more, but my scripts still need work in this area:

These two are merged:
1370602529,276,1,69
1370602529,84,7,158
This one is outside the merged sequences:
1370602529,38,21,5071

These two are merged:
1849563931,15472,9,1741
1849563931,70,15,1582
This one is outside the merged sequences:
1849563931,43,57,1

Here is the entire list (omitting the above triplets):

1000066283,157,6,1611
1000066283,80,33,1128
1001655821,20,67,692
1001655821,73,28,1192
1005523957,46,38,48
1005523957,99,44,1332
1006120013,167,6,165
1006120013,7,16,719
1008355279,103,8,1220
1008355279,86,41,38
1010085299,37,6,2502
1010085299,46,65,653
1012433291,101,4,1556
1012433291,30,37,2112
1013226583,6,23,544
1013226583,61,18,3451
1014907567,13,50,1573
1014907567,21,20,329
1027180397,34,71,160
1027180397,51,81,24
1035744623,109,38,628
1035744623,6,47,1523
1038824309,5,48,1350
1038824309,62,57,517
1040789047,10,47,500
1040789047,61,70,232
1051654267,19,22,2102
1051654267,2,468,58
1057464631,119,44,195
1057464631,87,38,1726
1061367701,10,77,3358
1061367701,11,68,2558
10655948837,19,56,310
10655948837,33,36,599
1068398117,113,30,200
1068398117,19,88,141
1068829721,11,68,3819
1068829721,34,49,281
10689931771,30,23,782
10689931771,46,13,392
1070974591,107,40,1628
1070974591,5,136,3285
1071660347,15,34,1721
1071660347,41,28,169
1073335301,50,18,8
1073335301,93,20,20
10749423863,11,116,3078
10749423863,30,5,1531
1078474589,10,37,1132
1078474589,19,76,1235
1088336201,26,73,395
1088336201,54,33,155
1097038783,2,501,61
1097038783,67,83,11
109964639887,193,22,263
109964639887,40,19,182
1104735523,3,106,342
1104735523,7,158,510
1108463827,43,20,227
1108463827,6,53,577
1108476167,11,88,15
1108476167,23,44,3475
1114626413,19,52,2714
1114626413,30,25,267
1117906721,109,36,220
1117906721,11,50,711
1118156813,46,27,334
1118156813,70,19,161
1121641319,10,37,2588
1121641319,57,48,200
1127225591,41,74,127
1127225591,51,58,1063
1128708899,119,20,94
1128708899,26,79,122
1129552253,19,18,510
1129552253,2,341,25
1130529443,26,51,719
1130529443,57,28,2387
1134318569,151,4,144
1134318569,95,26,161
1134335339,17,92,2030
1134335339,73,34,367
1134461173,18,79,74
1134461173,6,35,1513
1134797117,42,9,252
1134797117,7,98,969
1138574441,44,69,1203
1138574441,99,32,2256
1142266651,11,116,2980
1142266651,85,12,4627
1144636907,14,45,396
1144636907,3,146,561
1145898077,10,133,462
1145898077,70,29,1651
1146191047,31,28,66
1146191047,5,72,721
1150135219,11,29,4
1150135219,35,28,209
1153942483,3,211,7
1153942483,44,53,2201
1157788561,22,87,76
1157788561,46,33,1095
1158926663,26,31,550
1158926663,79,12,2168
1167137717,69,20,703
1167137717,79,8,528
1170750769,55,26,1389
1170750769,83,8,282
1180516231,29,15,1366
1180516231,7,104,955
1187125579,55,32,364
1187125579,65,52,79
1193918981,35,10,1480
1193918981,59,4,831
1194671551,191,34,223
1194671551,5,170,627
1195389149,17,22,608
1195389149,61,20,86
1195502977,19,32,1315
1195502977,6,133,785
1198306349,76,25,2555
1198306349,79,24,398
1205900173,12,69,1027
1205900173,38,29,755
1207589143,24,15,466
1207589143,3,58,3357
1221597011,37,22,1144
1221597011,5,82,750
1221597011,6,73,2552
12227903759,20,69,893
12227903759,54,23,1030
1229334677,35,50,540
1229334677,38,27,33
1234453553,3,214,2283
1234453553,94,37,271
1234704811,113,20,179
1234704811,28,43,135
1239548699,5,136,117
1239548699,52,27,797
1257729269,19,46,123
1257729269,77,8,421
1259803487,44,47,175
1259803487,6,73,1724
1264657091,14,71,1359
1264657091,94,45,628
1267449919,26,97,832
1267449919,43,8,2363
1267528987,3,34,252
1267528987,66,9,1037
1270550651,45,24,625
1270550651,76,19,783
12742648307,21,54,2903
12742648307,28,17,151
1274602453,69,56,810
1274602453,97,26,401
12755626837,10,17,754
12755626837,17,6,3969
1277097533,11,18,593
1277097533,74,57,1418
1278222611,39,22,290
1278222611,61,46,368
1279946509,14,89,1255
1279946509,41,24,750
1288234043,26,39,3163
1288234043,70,53,29
1291308467,19,76,2878
1291308467,7,48,1115
1293967729,19,42,92
1293967729,82,41,626
1294170701,30,25,1418
1294170701,37,22,3218
1298562247,29,16,3065
1298562247,60,18,3
1307652473,15,98,15
1307652473,44,35,2513
1309129933,3,68,1069
1309129933,48,31,714
1309588837,101,20,115
1309588837,14,77,208
1317570083,12,71,172
1317570083,29,40,273
1326169571,15,4,853
1326169571,39,32,1557
13273901521,42,65,81
13273901521,75,10,4528
1329031787,17,46,1751
1329031787,58,69,733
1331215709,44,27,657
1331215709,58,9,931
1331502647,24,61,92
1331502647,79,26,398
1334284499,68,45,388
1334284499,82,3,1467
1334602861,197,49,32
1334602861,75,16,852
1339215721,15,36,307
1339215721,69,24,316
13441505989,15,20,908
13441505989,17,20,1044
1349251559,17,92,1245
1349251559,61,68,170
1350352741,11,20,1203
1350352741,22,15,1475
1364492747,29,10,1071
1364492747,66,19,1327
1371644347,38,55,1181
1371644347,77,6,88
1373075419,42,23,661
1373075419,62,27,49
1375618577,5,136,1315
1375618577,51,32,633
1385682629,17,70,199
1385682629,51,74,1313
1388392903,20,25,1069
1388392903,58,11,169
1388696473,22,31,1809
1388696473,99,20,1224
1399913257,45,39,42
1399913257,6,11,298
1400714869,107,22,1039
1400714869,22,13,3897
1417019537,11,36,1691
1417019537,46,31,279
142754777149,101,32,504
142754777149,52,39,640
1428485593,13,16,376
1428485593,74,57,1188
1442122849,10,35,2506
1442122849,19,20,1399
1443633649,105,48,58
1443633649,109,10,386
1447677043,151,24,34
1447677043,28,31,1017
1453059701,35,10,2424
1453059701,57,38,792
1455497123,26,37,2577
1455497123,34,51,1425
1455649541,44,11,307
1455649541,86,15,304
1456319983,10,131,1394
1456319983,5,70,669
1458245561,60,63,96
1458245561,73,61,41
1458985963,3,190,1280
1458985963,59,24,523
1473659471,101,48,77
1473659471,15,34,1246
1476281803,127,42,939
1476281803,58,7,605
1494655627,10,123,334
1494655627,12,51,1186
1514980781,46,63,17
1514980781,99,12,887
1520335771,12,113,137
1520335771,19,105,80
1520459723,22,41,1246
1520459723,29,91,105
1539077593,41,52,232
1539077593,5,164,143
154198251007,17,76,702
154198251007,22,87,1336
1542335789,149,20,506
1542335789,79,10,228
1545539087,56,5,810
1545539087,94,13,2113
1573252489,31,86,120
1573252489,78,43,915
1575372067,10,121,1314
1575372067,35,32,1858
1576788293,54,61,1078
1576788293,55,30,800
1581829609,86,19,305
1581829609,97,40,2150
1592634581,35,10,58
1592634581,58,37,1275
1600807519,30,27,260
1600807519,6,89,821
1622062093,14,89,637
1622062093,46,58,43
1625043389,34,65,1601
1625043389,58,69,845
1639132051,2,422,11
1639132051,22,41,508
16409695231,38,21,2631
16409695231,6,7,253
1646665309,41,26,306
1646665309,99,12,334
1648370369,33,72,491
1648370369,70,62,7
1652665811,119,44,14
1652665811,26,73,1337
1661797037,22,41,396
1661797037,31,50,1756
1662398893,50,64,40
1662398893,68,31,1626
1668820973,12,97,23
1668820973,57,70,327
1671507989,70,29,341
1671507989,79,28,343
1672617637,15,114,298
1672617637,62,23,273
1691208683,55,12,1800
1691208683,93,6,1086
1691734123,109,8,205
1691734123,61,14,219
16955341633,12,23,488
16955341633,74,39,2353
1714204097,34,45,1669
1714204097,44,75,2169
1725886607,10,39,357
1725886607,15,98,1191
1728697067,14,17,563
1728697067,54,7,1236
1738718167,109,46,167
1738718167,23,18,1210
1743048779,10,77,1440
1743048779,3,110,956
1748701403,54,47,974
1748701403,76,11,2431
1751490079,3,152,892
1751490079,6,161,1959
1757771591,3,110,2457
1757771591,62,19,74
1758281221,62,11,1019
1758281221,7,18,472
1767263009,11,70,2398
1767263009,14,53,1440
1770121069,44,35,675
1770121069,57,8,2118
1782789427,33,24,636
1782789427,45,10,679
1786026601,101,22,403
1786026601,40,33,439
1789860791,29,36,654
1789860791,69,22,597
1800704767,127,8,121
1800704767,78,37,2122
1807530251,54,3,233
1807530251,96,11,583
1818419051,22,15,394
1818419051,3,70,1243
1821849553,67,38,1203
1821849553,83,6,1017
1823692711,20,27,1481
1823692711,86,5,1051
1848424423,14,29,365
1848424423,28,51,67
1888970921,17,34,428
1888970921,3,14,1694
1889889973,103,26,1378
1889889973,20,41,1
1897073869,12,37,336
1897073869,90,31,1881
1900569103,21,38,414
1900569103,5,74,2044
1909012393,2,195,46
1909012393,6,99,110
1909553381,167,20,429
1909553381,62,65,407
1912420217,101,36,1187
1912420217,7,132,2790
1930550801,30,3,1761
1930550801,53,58,215
1945784677,33,58,42
1945784677,75,26,582
1978558717,127,24,1546
1978558717,58,41,3703
19805197549,21,34,715
19805197549,28,71,128
1991389019,11,14,614
1991389019,6,33,373
1993048741,137,14,810
1993048741,35,56,526
19971331781,47,68,1399
19971331781,93,12,207

Of note, although I ran the primes against the entire set of bases, only 276 and 15472 appear in the results. And, they're only in the triplets above. I didn't notice any greater than 193 in the last listing. I also didn't notice any triplets that didn't involve merged sequences. There are some primes of greater than 10 digits in the listing.

[garambois]

Originally Posted by EdH

I have not solved the entire problem with merged duplication entirely, but I have come up with some results. I ran some work on all the primes from all the sequences from base 2 through base 200, that have the first digit of 1, against the entire set. In my attempt to cull those that appeared in merged sequences, I did find a couple triplets where a prime exists within merged sequences and outside of them. I suspect many more, but my scripts still need work in this area:
...

...

Here is the entire list (omitting the above triplets):

1000066283,157,6,1611
 ...
1221597011,37,22,1144
1221597011,5,82,750
1221597011,6,73,2552
...

Of note, although I ran the primes against the entire set of bases, only 276 and 15472 appear in the results. And, they're only in the triplets above. I didn't notice any greater than 193 in the last listing. I also didn't notice any triplets that didn't involve merged sequences. There are some primes of greater than 10 digits in the listing.

I did some quick checking.
I'll take a more detailed look over the next weekend.

It looks like there are also some 11-digit pairs as you already report.
This leads me to think that 12-digit pairs must be very rare, with the reduced number of sequences we have to do this research.
As for triplets, apart from the cases of mergers that you mention above, they must be very rare too.

Otherwise, I also note a triplet that does not involve a merged sequence for the prime 1221597011 (see above), but for the 37^22 sequence, at index 1144, the term is :
14995180448395695667673983126 = 2 - 3533 - 2122159701159877677281911
And so, 1221597011 is not a factor, but the part of writing a factor.

Thanks a lot Edwin !

[EdH]

Originally Posted by garambois

. . .Otherwise, I also note a triplet that does not involve a merged sequence for the prime 1221597011 (see above), but for the 37^22 sequence, at index 1144, the term is :
14995180448395695667673983126 = 2 - 3533 - 2122159701159877677281911
And so, 1221597011 is not a factor, but the part of writing a factor.

Thanks a lot Edwin !

I weeded out a bunch of those triplets, but I guess I missed one. . . That's the trouble with manually doing this. I still need to do better with programming/scripts. Thanks for keeping me in line.

[RichD]

I'll start initializing prime bases 223, 227 & 229.

[gd_barnes]

Originally Posted by garambois

For example, I also noticed yesterday that the prime 181 was missing from the list of all bases < 200 that are primes. I don't know why we had forgotten 181 ?
I will initialize it.

I had also missed this.

I am currently in the process of initializing just the same-parity exponents on bases 179 and 181 to add some interesting bases for the easier sequences effort. I started yesterday and it did not appear that those had been worked on because the elves were having to work to terminate the smaller same-parity exponents when I pulled them up.

I am currently up to exponent 55 on both bases. I have a batch job set up to initialize all of them up to a starting sequence of 160 digits...exponent 71. For now, the job stops any base that has a hard cofactor (fully ECM'd) of >= 102 digits. I will then run back through and run any remaining exponents until they hit a cofactor of >= 110 digits.

Hopefully this is OK.

[RichD]

Originally Posted by gd_barnes

I am currently in the process of initializing just the same-parity exponents on bases 179 and 181 to add some interesting bases for the easier sequences effort. I started yesterday and it did not appear that those had been worked on because the elves were having to work to terminate the smaller same-parity exponents when I pulled them up.

I am currently up to exponent 55 on both bases. I have a batch job set up to initialize all of them up to a starting sequence of 160 digits...exponent 71. For now, the job stops any base that has a hard cofactor (fully ECM'd) of >= 102 digits. I will then run back through and run any remaining exponents until they hit a cofactor of >= 110 digits.

Hopefully this is OK.

Both bases are currently assigned according to the reservation table.

[garambois]

Originally Posted by gd_barnes

I had also missed this.

I am currently in the process of initializing just the same-parity exponents on bases 179 and 181 to add some interesting bases for the easier sequences effort. I started yesterday and it did not appear that those had been worked on because the elves were having to work to terminate the smaller same-parity exponents when I pulled them up.

I am currently up to exponent 55 on both bases. I have a batch job set up to initialize all of them up to a starting sequence of 160 digits...exponent 71. For now, the job stops any base that has a hard cofactor (fully ECM'd) of >= 102 digits. I will then run back through and run any remaining exponents until they hit a cofactor of >= 110 digits.

Hopefully this is OK.

Attention, I am calculating myself the sequences of bases 179 and 181, as RichD says.
It's just that I hadn't reported the results to FactorDB yet.
I am afraid that we have done the same calculations twice.
Please look at the last line of the table under the heading "Navigation" on the main project page : bases 179 and 181 were reserved for me.
Should I stop my calculations on these two bases if you have also done them ?
Have you also done calculations for the exponents of parity opposite to the base ?

[garambois]

Originally Posted by RichD

I'll start initializing prime bases 223, 227 & 229.

OK, many thanks !
It will extend my area of analysis a bit this summer if we integrated bases that are exhaustively prime numbers further than 200.