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-   -   Aliquot sequences that start on the integer powers n^i (https://www.mersenneforum.org/showthread.php?t=23612)

RichD 2021-10-12 23:47

Base 66 can be added at the next update.

yoyo 2021-10-14 15:16

I'll take bases 51, 52, 54, 55, 56, 57, 59.

garambois 2021-10-15 17:34

[QUOTE=RichD;590345]Base 66 can be added at the next update.[/QUOTE]

[QUOTE=yoyo;590549]I'll take bases 51, 52, 54, 55, 56, 57, 59.[/QUOTE]


Thank you for this work and for these new reservations.
The next update will take place around the next weekend (October 23, 24).
I'm thinking of doing the first tests for a new data analysis idea in the next few weeks.
I will keep you posted if I find something interesting.
This new idea is for bases below 100.
Don't forget to enter all your latest results on factordb by Tuesday, October 26, when I will start scanning the sequences.

garambois 2021-10-25 08:46

Page updated.
Many thanks to all for your help !

[B]Added base : 66.[/B]
[B]Still no non-trivial end of sequence for bases 276, 552.[/B]
[B]Yoyo found a non-trivial ending on a cycle for 52^55.[/B]

For your information, tomorrow, Tuesday, October 26, I will be scanning FactorDB for all bases less than or equal to 100 to do some new data analysis based on a new idea.

RichD 2021-10-25 12:13

Base 67 can be added at the next update. Many terms already in FDB.

Base 68 can be added. Not quite done but all indices have been touched with many moved along.

garambois 2021-10-26 06:29

Thank you very much RichD.
I will add these two bases to my FDB scan tonight at 6 PM, GMT.

garambois 2021-10-31 11:11

1 Attachment(s)
[SIZE=3][B]Here I would like to present my new idea to analyze the data of our project differently.[/B][/SIZE]

After a few days of reflection, I came to the conclusion that the most effective way was to do this.
For each prime number P, I draw a set of points in a system of coordinate axes.
On the x-axis, we have the base and on the y-axis, we have the exponent.
We plot the points corresponding to all the sequences that end with this prime number P.
The goal is to find points aligned on a line or a set of points that would appear to be on a regular curve, whose equation could then be determined.
Here is below the result of my analysis following the scan of FDB of October 26 :

[CODE]Points for the prime 2
[(2, 1)]
Points for the prime 3
[(2, 2), (2, 4), (2, 55), (2, 164), (2, 305), (2, 317), (3, 1), (3, 2), (3, 5), (3, 247), (5, 38), (6, 152), (7, 4), (7, 77), (10, 124), (11, 2), (11, 15), (12, 1), (12, 2), (13, 15), (14, 76), (14, 80), (15, 1), (19, 15), (20, 2), (20, 8), (21, 21), (21, 55), (22, 80), (23, 3), (26, 1), (29, 2), (29, 69), (30, 1), (30, 82), (30, 92), (31, 79), (33, 1), (35, 49), (37, 11), (38, 11), (38, 30), (40, 14), (41, 2), (42, 1), (43, 15), (44, 58), (45, 1), (45, 34), (46, 1), (47, 13), (50, 73), (52, 1), (52, 34), (53, 2), (53, 27), (53, 30), (54, 1), (54, 10), (55, 53), (58, 18), (59, 49), (66, 1), (72, 1), (74, 6), (74, 20), (75, 45), (79, 97), (98, 45)]
Points for the prime 5
[(5, 1)]
Points for the prime 7
[(2, 3), (2, 10), (2, 12), (2, 141), (2, 278), (2, 387), (2, 421), (3, 6), (3, 8), (3, 118), (3, 198), (3, 305), (7, 1), (7, 2), (7, 8), (7, 127), (10, 1), (11, 5), (12, 21), (13, 2), (13, 87), (14, 1), (14, 19), (14, 21), (15, 10), (17, 24), (17, 91), (18, 13), (18, 52), (18, 70), (19, 2), (20, 1), (21, 8), (21, 17), (22, 1), (22, 19), (23, 11), (26, 3), (26, 19), (26, 50), (26, 80), (28, 9), (28, 47), (30, 52), (30, 70), (34, 1), (34, 7), (34, 9), (35, 11), (37, 2), (38, 1), (38, 13), (40, 2), (41, 49), (42, 2), (43, 20), (45, 4), (45, 55), (46, 33), (47, 8), (47, 42), (48, 61), (50, 17), (53, 8), (56, 38), (58, 5), (58, 27), (61, 2), (62, 1), (66, 5), (68, 10), (72, 25), (74, 11), (75, 1), (75, 63), (79, 7)]
Points for the prime 11
[(2, 60), (2, 316), (2, 480), (2, 499), (3, 15), (3, 189), (3, 303), (5, 15), (7, 143), (10, 20), (11, 1), (11, 137), (13, 31), (14, 14), (14, 28), (17, 2), (18, 1), (18, 55), (18, 76), (21, 1), (21, 47), (24, 2), (26, 36), (35, 83), (37, 50), (47, 47), (51, 1), (53, 15), (53, 67), (58, 2), (58, 24), (65, 2), (65, 33), (67, 3), (72, 4), (72, 58), (72, 63), (79, 3), (98, 2)]
Points for the prime 13
[(2, 358), (3, 3), (3, 31), (3, 67), (5, 9), (11, 3), (11, 27), (13, 1), (20, 72), (23, 49), (23, 77), (24, 70), (35, 1), (38, 86), (43, 3), (43, 17), (50, 62), (56, 8), (62, 2), (75, 13), (79, 23)]
Points for the prime 17
[(3, 63), (6, 2), (7, 55), (13, 21), (15, 79), (17, 1), (19, 97), (23, 2), (24, 1), (24, 4), (38, 42), (39, 1), (45, 5), (46, 10), (51, 31), (55, 1), (58, 70), (60, 28)]
Points for the prime 19
[(2, 39), (2, 76), (2, 190), (2, 219), (2, 505), (3, 275), (5, 233), (10, 2), (10, 12), (10, 44), (12, 4), (13, 3), (13, 141), (15, 21), (19, 1), (22, 32), (40, 6), (41, 7), (42, 10), (42, 22), (45, 15), (55, 29), (60, 30), (65, 1), (72, 74), (79, 13), (98, 6)]
Points for the prime 23
[(3, 12), (3, 181), (7, 3), (7, 9), (11, 127), (17, 79), (18, 6), (18, 17), (18, 64), (21, 61), (23, 1), (40, 36), (44, 68), (47, 3), (55, 15), (57, 1), (57, 17), (65, 3), (74, 64), (79, 49)]
Points for the prime 29
[(5, 41), (14, 92), (18, 50), (18, 116), (22, 8), (26, 68), (29, 1), (46, 36), (47, 39), (48, 4), (61, 5), (63, 67), (72, 43)]
Points for the prime 31
[(2, 5), (2, 101), (2, 146), (3, 169), (5, 3), (5, 161), (6, 17), (11, 38), (11, 93), (11, 107), (12, 14), (19, 35), (19, 37), (31, 1), (31, 2), (31, 11), (40, 30), (44, 30), (56, 52), (58, 1), (61, 55), (67, 2), (68, 1), (74, 21), (98, 3)]
Points for the prime 37
[(2, 68), (2, 125), (2, 243), (3, 90), (5, 4), (7, 6), (7, 79), (10, 3), (10, 5), (10, 74), (11, 40), (12, 98), (14, 2), (14, 35), (22, 7), (22, 21), (22, 23), (22, 74), (23, 10), (24, 6), (26, 31), (28, 3), (28, 11), (29, 50), (35, 2), (37, 1), (42, 16), (44, 3), (44, 19), (45, 11), (47, 16), (51, 2), (53, 87), (55, 43), (63, 3), (74, 5), (75, 8), (79, 6)]
Points for the prime 41
[(2, 6), (2, 8), (2, 23), (2, 47), (2, 112), (2, 117), (2, 281), (2, 373), (2, 405), (2, 411), (6, 5), (6, 13), (7, 11), (11, 14), (11, 57), (12, 32), (12, 56), (12, 119), (15, 2), (15, 6), (17, 65), (18, 39), (20, 3), (24, 18), (33, 2), (35, 8), (35, 34), (37, 20), (37, 73), (38, 54), (40, 11), (40, 20), (40, 21), (40, 52), (41, 1), (42, 5), (43, 25), (43, 81), (44, 49), (45, 8), (45, 22), (45, 36), (47, 2), (48, 1), (48, 3), (50, 16), (51, 18), (52, 25), (55, 21), (56, 1), (57, 2), (59, 39), (61, 9), (62, 3), (62, 13), (62, 40), (63, 1), (65, 4), (66, 25), (75, 2), (79, 2), (79, 4), (79, 17), (98, 10), (98, 41)]
Points for the prime 43
[(2, 9), (2, 62), (2, 210), (2, 271), (2, 510), (3, 4), (3, 22), (3, 80), (3, 86), (6, 26), (6, 32), (6, 77), (7, 16), (7, 28), (7, 51), (10, 42), (11, 13), (11, 28), (11, 56), (13, 4), (13, 8), (13, 40), (13, 95), (13, 98), (14, 5), (14, 23), (14, 25), (15, 14), (17, 12), (17, 16), (17, 101), (18, 34), (19, 4), (19, 55), (20, 21), (20, 70), (21, 2), (21, 6), (21, 98), (22, 2), (23, 4), (24, 10), (24, 19), (24, 31), (24, 36), (26, 6), (26, 38), (26, 78), (26, 88), (28, 13), (28, 18), (29, 3), (29, 59), (31, 14), (31, 26), (33, 3), (33, 23), (33, 45), (34, 3), (34, 28), (34, 44), (35, 4), (37, 97), (38, 38), (39, 6), (40, 1), (40, 35), (42, 13), (43, 1), (43, 2), (43, 55), (44, 1), (45, 2), (46, 3), (46, 5), (47, 17), (48, 9), (50, 1), (51, 5), (51, 13), (51, 50), (52, 9), (53, 4), (57, 26), (59, 2), (59, 6), (59, 18), (60, 1), (60, 6), (62, 8), (62, 9), (62, 50), (63, 6), (65, 12), (65, 32), (66, 21), (66, 56), (67, 10), (67, 63), (68, 14), (68, 18), (74, 1), (75, 6), (75, 11), (75, 46), (75, 73), (79, 30), (79, 31), (79, 56), (98, 5), (98, 29)]
Points for the prime 47
[(2, 453), (3, 9), (28, 34), (30, 10), (40, 22), (47, 1), (65, 47), (98, 75)]
Points for the prime 53
[(2, 20), (2, 78), (2, 214), (2, 347), (2, 450), (48, 50), (48, 70), (53, 1), (72, 18), (75, 31)]
Points for the prime 59
[(3, 94), (6, 128), (7, 14), (7, 24), (7, 64), (10, 7), (10, 23), (11, 8), (11, 58), (11, 80), (13, 6), (13, 38), (14, 9), (14, 49), (15, 36), (17, 64), (19, 6), (19, 25), (20, 59), (20, 71), (24, 9), (24, 13), (26, 59), (26, 71), (28, 7), (28, 23), (29, 4), (29, 36), (31, 8), (31, 10), (33, 4), (33, 6), (35, 6), (38, 77), (39, 2), (39, 12), (42, 3), (44, 5), (45, 6), (45, 28), (45, 30), (51, 32), (52, 3), (52, 11), (55, 4), (57, 4), (58, 17), (59, 1), (60, 3), (60, 17), (61, 12), (65, 50), (66, 17), (74, 33), (75, 18), (75, 38)]
Points for the prime 61
[(30, 90), (39, 3), (43, 23), (45, 7), (52, 6), (52, 36), (60, 9), (61, 1), (67, 61), (75, 43)]
Points for the prime 67
[(2, 147), (14, 6), (19, 53), (45, 47), (57, 13), (67, 1)]
Points for the prime 71
[(2, 43), (2, 177), (6, 14), (12, 58), (13, 53), (13, 113), (18, 77), (21, 11), (22, 36), (26, 66), (50, 2), (61, 11), (63, 33), (66, 70)]
Points for the prime 73
[(2, 67), (3, 98), (5, 10), (6, 3), (7, 133), (10, 11), (10, 13), (12, 3), (17, 48), (20, 4), (22, 94), (23, 6), (28, 5), (31, 3), (34, 2), (34, 5), (43, 14), (46, 7), (57, 47), (60, 62), (63, 14), (65, 57), (74, 74), (98, 1)]
Points for the prime 79
[(3, 113), (6, 4), (17, 39), (34, 56), (50, 6), (79, 1)]
Points for the prime 83
[(2, 376), (5, 5), (5, 36), (5, 57), (56, 44)]
Points for the prime 89
[(2, 32), (2, 82), (3, 111), (7, 101), (37, 23), (47, 31), (57, 5), (68, 12)]
Points for the prime 97
[(2, 51), (2, 73), (7, 43), (19, 38), (38, 34), (43, 57), (68, 2), (98, 80)][/CODE]The result has been voluntarily limited to all bases < 100 for all primes < 100.
Indeed, it is only in the domain [0,100] for the bases that we have a sufficient density of calculated bases.
But maybe this is a mistake and maybe it would have been better to do this work for all bases up to 1000 ?

[U]How to read this data ?[/U]

[B]For example, for the prime number P=2[/B], there is only one sequence starting on a power of an integer in our project that ends with the prime P=2 : it is 2^1.
So there is only one point to plot for the prime number P=2, it is the point with coordinates x=2 and y=1.
But there is no interest in plotting only one point !

[B]But for the prime number P=3[/B], this is much more interesting.
For the prime number P=3, there are only 69 sequences that end with P=3.
These are : 2^2, 2^4, 2^55, 2^164, 2^305, 2^317, 3^1, 3^2, 3^5, 3^247, 5^38...
In our axis system, we can therefore plot 69 points whose coordinates are : (2,2), (2,4), (2,55), (2,164), (2,305)...

Same for the other bases.

I have attached to this post all the drawings of the point sets for each prime number < 100.
Unfortunately, in none of these drawings can I visually locate a regular line or curve that might be formed by some of these points.
Perhaps we need to analyze this data in a different way than simply placing these points in a system of axes and just looking by eye if there is nothing ?
But personally, I don't know how to attack this analysis other than by visualizing the positions of the points on an axis system.
And maybe there is nothing to see or maybe we don't have enough points ?

garambois 2021-11-10 18:01

Page updated.
Many thanks to all for your help !

[B]Added bases : 67, 68.[/B]
[B]Still no non-trivial end of sequence for bases 276, 552.[/B]

Our project now has 111 bases and 10162 sequences !

Nescro 2021-12-05 04:18

Base 69
 
While not fully complete, I think base 69 can potentially be added in the next update.

unconnected 2021-12-05 07:10

79^93 is finished, the last remaining odd number below 100 is 79^99.

garambois 2021-12-05 09:14

[QUOTE=Nescro;594497]While not fully complete, I think base 69 can potentially be added in the next update.[/QUOTE]

Many thanks, I will add base 69 in the next full update on the weekend of December 18 and 19.
I also hope that by then, some friends of the Alliance Francophone will do some calculations on the BOINC "yafu" project.
That should make the statistics go up a bit.


[QUOTE=unconnected;594502]79^93 is finished, the last remaining odd number below 100 is 79^99.[/QUOTE]

Many thanks.
These are large calculations, as they are sequences that end but begin with numbers over 165 digits.
79^99 even starts with a 177 digit number and the cofactor of index 28 seems recalcitrant !


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