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

√Čric 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.