10*987/6+54/3*21

Happy New Year!

Porch visitors

furries

The fire is so delightful

let it snow, let it blow, let it go

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 effort:

1671887:   128018188027

6264400:   918217008016

6524469: >2000000000000

7404616:  3306609997544

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:

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.