Friday, February 03, 2023

Lake effect

click to enlarge
It is sunny against a mostly blue sky backdrop outside but, as predicted, very cold. I did not even take Bodie for his walk this morning! My GOES 16 screen grab (above) shows snow streamers coming off the Great Lakes (left) with the prevailing northwesterly winds (below). The situation is even more pronounced off the coast of Maine (far right) where the lake effect is an ocean effect.
click to enlarge

Sunday, January 29, 2023

Wednesday, January 25, 2023

A million-digit Leyland prime (end of run)

manual distribution worksheet for 59536 primality tests

An hour ago I completed the primality check of 59536 Leyland-prime candidates that I started on 2 May 2022. My 27 May 2022 reality check explains why it took way longer (just short of nine months) than I had originally planned. Preceding the primality testing was another month (or so) sieving the original Leyland numbers file (so as to exclude divisibility by primes up to 2*10^11), so ten months altogether, now done.

What have I discovered? Of Leyland numbers with at least one million decimal digits, but fewer than one million one hundred decimal digits, there is only one prime. That prime was discovered by Gabor Levai long before I got to it. I saved all of my primality-test output where the 59536 entries are listed smallest to largest. If you want to see the one prime, search for "PRP".

The execution times vary wildly (due to processor circumstances) with an average of 9.45 hours per test. That would work out to 64 years if I hadn't been able to multi-process. I know now how to keep the execution times to 6 hours or less per test but that means running fewer processes per machine. Still, I might be able to shave a couple of months off the total time required for the next run.

Thursday, January 12, 2023

A digit-spine sequence

Éric Angelini presented a proposal for three "digit-spine" sequences on his blog here, as well as to the MathFun community. I decided to take on the first one:

s = 1, 10, 2, 0, 3, 26, 9, 119, 532, 4, 6, 896, 118, 34, 15, ...

p = 2, 11, 2, 2, 3, 23, 7, 113, 523, 3, 5, 887, 113, 31, 13, ...

d = 1,  1, 0, 2, 0,  3, 2,   6,   9, 1, 1,   9,   5,  3,  2, ...

Our sought-after sequence is 's'. This is followed by 'p', a sequence of examples (for each, there might be a second such) of primes that are as close to the terms of 's' as possible. Finally 'd', the absolute differences between the respective  's' and 'p' values. Determining 's' is that it must be the lexicographically earliest sequence of distinct nonnegative integers such that its digit-spine (the sequential digits) as well as the digit-spine of 'd' are identical.

For example, the third 'd' being 0 is a consequence of the zero in 10, the second term of 's'. This forces the third term of 's' to be the smallest prime (as it had not already been used). And so on. There will always be integers in 'd' that inform our upcoming values in 's'. And the digits of those values get added to the growing end of our 'd' sequence.

So far, all of our 'd' values are single digits. But there will come a point when concatenating a digit with the next one in line allows for a smaller 's' term than what is available for the single digit. This first happens at term #1622 where a d = 1 dictates s = 1010. The next d digit is 0. Concatenating that 0 with the previous 1 gives us d = 10 at #1622 and this allows the smaller s = 897. This makes me want to know at which term the first three-digit d appears.

Maximilian Hasler helped Éric get the sequence into the OEIS. I have an indexed listing up to one million terms here.

Monday, December 19, 2022

Binary complement sequences

On Friday, Joshua Searle posted to the Sequence Fanatics Discussion list a neat procedure: take the binary complement of an integer multiplied by 3. Iterate. For example, starting with 3 we get the binary of 9 (1001), the complement of which (0110) is 6. Continuing, from 6 we get the binary of 18 (10010), the complement of which (01101) is 13. Arriving at zero, we stop.

 0 3
 1 6
 2 13
 3 24
 4 55
 5 90
 6 241
 7 300
 8 123
 9 142
10 85
11 0

Eleven steps to get to zero. The largest integer reached is 300 at step 7. We can shorthand the sequence data for 3 with (11,7,300) [steps to reach zero, steps to reach a maximum, the maximum]. Here are the statistics for integer starts up to 28:

 0 (0,0,0)
 1 (1,0,1)
 2 (2,0,2)
 3 (11,7,300)
 4 (12,8,300)
 5 (1,0,5)
 6 (10,6,300)
 7 (3,1,10)
 8 (4,2,10)
 9 (13,9,300)
10 (2,0,10)
11 (19,15,300)
12 (80,28,328536)
13 (9,5,300)
14 (2,1,21)
15 (15,11,300)
16 (16,12,300)
17 (81,29,328536)
18 (14,10,300)
19 (11,7,300)
20 (12,8,300)
21 (1,0,21)
22 (6,2,72)
23 (83,31,328536)
24 (8,4,300)
25 (73,21,328536)
26 (22,5,661)
27 (79,27,328536)
28 (7572,2962,123130640068522377168864228132316865867184046004226894)

Note the large number of steps and maximum in the last one. Somewhat more surprising are subsequent records. Do all integer starts eventually reach zero? Tom Duff has provided a list of the progressive record number of steps to get to zero:

1: 1
2: 2
3: 11
4: 12
9: 13
11: 19
12: 80
17: 81
23: 83
28: 7572
33: 7573
74: 7574
86: 7578
180: 7580
227: 664475
350: 664882
821: 3180929
3822: 3180930
4187: 3180931
5561: 3181981
6380: 3181988
6398: 3182002
22174: 3182226
22246: 120796790
26494: 556068798
34859: 556068799
49827: 556068871
70772: 556068872
103721: 572086553
104282: 572086610
204953: 1246707529
213884: 1246707552
225095: 1246707555
407354: 1246707602
425720: 87037147316

Update: Tim Peters has been working on extending the outcome of start numbers up to ten million. Here are a few results from his ongoing effort:

1671887:  128018188027
6264400: >383000000000
6524469: >327000000000
7011851:  225197172159
7404616: >370000000000

Saturday, December 10, 2022

Mobile upgrade

iPhone 6 on warp drive

Catherine's almost-seven-year-old iPhone 6 was dying/dead so I ordered a replacement.


Starting it up went well enough although I did spend some fifteen minutes prior looking for a paper clip. Only later did I discover the iPhone 13's accompanying SIM-tray ejector tool:

Thursday, December 08, 2022

Products with embedded indices

Éric Angelini did a "smallest multiplication" bit yesterday that I felt was worth extending.

     0 0 = 0 * 1
     1 10 = 2 * 5
     2 12 = 3 * 4
     3 132 = 6 * 22
     4 84 = 7 * 12
     5 152 = 8 * 19
     6 126 = 9 * 14
     7 170 = 10 * 17
     8 198 = 11 * 18
     9 195 = 13 * 15
    10 1008 = 16 * 63
    11 1100 = 20 * 55
    12 1218 = 21 * 58
    13 713 = 23 * 31
    14 1416 = 24 * 59
    15 1150 = 25 * 46
    16 1612 = 26 * 62
    17 1728 = 27 * 64
    18 1820 = 28 * 65
    19 1914 = 29 * 66
    20 1020 = 30 * 34
    21 1216 = 32 * 38
    22 1221 = 33 * 37
    23 2345 = 35 * 67
    24 2448 = 36 * 68
    25 2925 = 39 * 75
    26 12600 = 40 * 315
    27 1927 = 41 * 47
    28 2898 = 42 * 69
   ...

The column of indices on the far left is shown embedded (in bold) in their adjacent products. The constraint on the multiplier and multiplicand is that they must be distinct nonnegative integers with the multiplier the smallest such not yet used and the multiplicand the smallest such that yields the embedded index. A chart extending the products is here.