Thursday, June 13, 2024

Equal integer-name rank-products

As I'm looking through the extended list for my previous-post idea (now OEIS A373391), I will occasionally come across occurrences where the integer-name split fortuitously creates a left side and a right side where both are valid integer names. For example:

85740 = eightyfivethousand|sevenhundredforty => 85000 & 740

This gave me the idea to tease out all such number pairs (less than one million):

(147,741) & (1085,5081,81005,85001)
(247,742) & (2085,5082,82005,85002)
(347,743) & (3085,5083,83005,85003)
(357,753) & (10035,35010)
(447,744) & (4085,5084,84005,85004)
(547,745) & (5085,85005)
(647,746) & (5086,6085,85006,86005)
(740) & (5080,80005,85000)
(747) & (5087,7085,85007,87005)
(748,847) & (5088,8085,85008,88005)
(749,947) & (5089,9085,85009,89005)
(815) & (930)
(1031,31001) & (18040,40018)
(1131,31101,101031,131001) & (18140,40118,118040,140018)
...

The full list is here. There are 586 solutions of which 569 are pairs, 16 are triples, and 1 is a quadruple. Notice that for almost all entries there are variants that will not change the rank-product. In my above example, 740 allows 5080 and 80005 in addition to 85000 as its partners (fivethousandeighty, eightythousandfive, eightyfivethousand are letter-equivalent). 747 has four letter-equivalent partners. But 815 (no variants) has only 930 (no variants), which appears to make it unique in that regard in our number range:

815 = eighthundredfifteen => (5*9*7*8*20*8*21*14*4*18*5*4*6*9*6*20*5*5*14) = 387144769536000000
930 = ninehundredthirty   => (14*9*14*5*8*21*14*4*18*5*4*20*8*9*18*20*25)  = 
387144769536000000

Since all of these variants are a little obtrusive and don't contribute a lot to our display, I've created a new solutions list that shows only the first (smallest) integer of every variant. Now the quadruple at 1138 is easily spotted.

Here's a nice large-number example of the genre: 10^15+14 (onequadrillionfourteen) has the same rank-product as 10^147 (oneoctoquadragintillion).

Saturday, June 01, 2024

Equal products in split English integer names

    1           3960  threethousandni|nehundredsixty
    2          13986  thirteenthousandni|nehundredeightysix
    3          15368  fifteenthousandthre|ehundredsixtyeight
    4          80547  eightythousandfive|hundredfortyseven
    5          85740  eightyfivethousand|sevenhundredforty
    6         111789  onehundredeleventhousan|dsevenhundredeightynine
    7         111987  onehundredeleventhousan|dninehundredeightyseven
    8         386048  threehundredeightysi|xthousandfortyeight
    9         408649  fourhundredeightthous|andsixhundredfortynine
                ==    fourhundredeightthousa|ndsixhundredfortynine
   10         408946  fourhundredeightthous|andninehundredfortysix
                ==    fourhundredeightthousa|ndninehundredfortysix
   11         410699  fourhundredtenthousan|dsixhundredninetynine
   12         410969  fourhundredtenthousan|dninehundredsixtynine
   13         410996  fourhundredtenthousan|dninehundredninetysix
   14         486014  fourhundredeightys|ixthousandfourteen
   15         519487  fivehundrednineteenthous|andfourhundredeightyseven
                ==    fivehundrednineteenthousa|ndfourhundredeightyseven
   16         519784  fivehundrednineteenthous|andsevenhundredeightyfour
                ==    fivehundrednineteenthousa|ndsevenhundredeightyfour
   17         609408  sixhundredninethou|sandfourhundredeight
   18         609804  sixhundredninethou|sandeighthundredfour
   19         615430  sixhundredfifteenthou|sandfourhundredthirty
   20         619814  sixhundrednineteenthou|sandeighthundredfourteen
   21         629428  sixhundredtwentyninethou|sandfourhundredtwentyeight
   22         629824  sixhundredtwentyninethou|sandeighthundredtwentyfour
   23         639438  sixhundredthirtyninethou|sandfourhundredthirtyeight
   24         639834  sixhundredthirtyninethou|sandeighthundredthirtyfour
   25         649448  sixhundredfortyninethou|sandfourhundredfortyeight
   26         649844  sixhundredfortyninethou|sandeighthundredfortyfour
   27         659458  sixhundredfiftyninethou|sandfourhundredfiftyeight
   28         659854  sixhundredfiftyninethou|sandeighthundredfiftyfour
   29         669468  sixhundredsixtyninethou|sandfourhundredsixtyeight
   30         669864  sixhundredsixtyninethou|sandeighthundredsixtyfour
   31         679478  sixhundredseventyninethou|sandfourhundredseventyeight
   32         679874  sixhundredseventyninethou|sandeighthundredseventyfour
   33         686016  sixhundredeightys|ixthousandsixteen
   34         688050  sixhundredeightye|ightthousandfifty
   35         689400  sixhundredeightynin|ethousandfourhundred
   36         689488  sixhundredeightyninethou|sandfourhundredeightyeight
   37         689884  sixhundredeightyninethou|sandeighthundredeightyfour
   38         696468  sixhundredninetysixthou|sandfourhundredsixtyeight
   39         696864  sixhundredninetysixthou|sandeighthundredsixtyfour
   40         697478  sixhundredninetyseventhou|sandfourhundredseventyeight
   41         697874  sixhundredninetyseventhou|sandeighthundredseventyfour
   42         699314  sixhundredninetyninetho|usandthreehundredfourteen
   43         699498  sixhundredninetyninethou|sandfourhundredninetyeight
   44         699894  sixhundredninetyninethou|sandeighthundredninetyfour
   45         738045  sevenhundredthirtyei|ghtthousandfortyfive
   46         740085  sevenhundredforty|thousandeightyfive
   47         745080  sevenhundredforty|fivethousandeighty
   48         769478  sevenhundredsixtyninethou|sandfourhundredseventyeight
   49         769874  sevenhundredsixtyninethou|sandeighthundredseventyfour
   50         786017  sevenhundredeightys|ixthousandseventeen
   51         787016  sevenhundredeightys|eventhousandsixteen
   52         796478  sevenhundredninetysixthou|sandfourhundredseventyeight
   53         796874  sevenhundredninetysixthou|sandeighthundredseventyfour
   54         804649  eighthundredfourthous|andsixhundredfortynine
                ==    eighthundredfourthousa|ndsixhundredfortynine
   55         804946  eighthundredfourthous|andninehundredfortysix
                ==    eighthundredfourthousa|ndninehundredfortysix
   56         847035  eighthundredfortysev|enthousandthirtyfive
   57         875039  eighthundredseventyfi|vethousandthirtynine
   58         906408  ninehundredsixthou|sandfourhundredeight
   59         906804  ninehundredsixthou|sandeighthundredfour
   60         916814  ninehundredsixteenthou|sandeighthundredfourteen
   61         926428  ninehundredtwentysixthou|sandfourhundredtwentyeight
   62         926824  ninehundredtwentysixthou|sandeighthundredtwentyfour
   63         936438  ninehundredthirtysixthou|sandfourhundredthirtyeight
   64         936834  ninehundredthirtysixthou|sandeighthundredthirtyfour
   65         946448  ninehundredfortysixthou|sandfourhundredfortyeight
   66         946844  ninehundredfortysixthou|sandeighthundredfortyfour
   67         956458  ninehundredfiftysixthou|sandfourhundredfiftyeight
   68         956854  ninehundredfiftysixthou|sandeighthundredfiftyfour
   69         966468  ninehundredsixtysixthou|sandfourhundredsixtyeight
   70         966864  ninehundredsixtysixthou|sandeighthundredsixtyfour
   71         967478  ninehundredsixtyseventhou|sandfourhundredseventyeight
   72         967874  ninehundredsixtyseventhou|sandeighthundredseventyfour
   73         969314  ninehundredsixtyninetho|usandthreehundredfourteen
   74         969498  ninehundredsixtyninethou|sandfourhundredninetyeight
   75         969894  ninehundredsixtyninethou|sandeighthundredninetyfour
   76         976478  ninehundredseventysixthou|sandfourhundredseventyeight
   77         976874  ninehundredseventysixthou|sandeighthundredseventyfour
   78         986019  ninehundredeightys|ixthousandnineteen
   79         986488  ninehundredeightysixthou|sandfourhundredeightyeight
   80         986884  ninehundredeightysixthou|sandeighthundredeightyfour
   81         996314  ninehundredninetysixtho|usandthreehundredfourteen
   82         996498  ninehundredninetysixthou|sandfourhundredninetyeight
   83         996894  ninehundredninetysixthou|sandeighthundredninetyfour
   84     1000107588  onebilliononehundredseventh|ousandfivehundredeightyeight
   85     1000107885  onebilliononehundredseventh|ousandeighthundredeightyfive
   86     1000307155  onebillionthreehundredseven|thousandonehundredfiftyfive
   87     1000307551  onebillionthreehundredseven|thousandfivehundredfiftyone
   88     1000700135  onebillionsevenhundredtho|usandonehundredthirtyfive
   89     1000700531  onebillionsevenhundredtho|usandfivehundredthirtyone
   90     1000701588  onebillionsevenhundredoneth|ousandfivehundredeightyeight
   91     1000701885  onebillionsevenhundredoneth|ousandeighthundredeightyfive
   92     1000703155  onebillionsevenhundredthree|thousandonehundredfiftyfive
   93     1000703551  onebillionsevenhundredthree|thousandfivehundredfiftyone
   94     1000710185  onebillionsevenhundredtent|housandonehundredeightyfive
   95     1000710581  onebillionsevenhundredtent|housandfivehundredeightyone
   96     1000722509  onebillionsevenhundredtwent|ytwothousandfivehundrednine
   97     1000722905  onebillionsevenhundredtwent|ytwothousandninehundredfive
   98     1000725209  onebillionsevenhundredtwent|yfivethousandtwohundrednine
   99     1000725902  onebillionsevenhundredtwent|yfivethousandninehundredtwo
  100     1000729205  onebillionsevenhundredtwent|yninethousandtwohundredfive
  101     1000729502  onebillionsevenhundredtwent|yninethousandfivehundredtwo
  102     1000733155  onebillionsevenhundredthirtyt|hreethousandonehundredfiftyfive
  103     1000733551  onebillionsevenhundredthirtyt|hreethousandfivehundredfiftyone
  104     1000817155  onebillioneighthundredsevente|enthousandonehundredfiftyfive
  105     1000817551  onebillioneighthundredsevente|enthousandfivehundredfiftyone
  106     1000879185  onebillioneighthundredseventyn|inethousandonehundredeightyfive
  107     1000879581  onebillioneighthundredseventyn|inethousandfivehundredeightyone
  108     2000307255  twobillionthreehundredseven|thousandtwohundredfiftyfive
  109     2000307552  twobillionthreehundredseven|thousandfivehundredfiftytwo
  110     2000700235  twobillionsevenhundredtho|usandtwohundredthirtyfive
  111     2000700532  twobillionsevenhundredtho|usandfivehundredthirtytwo
  112     2000703255  twobillionsevenhundredthree|thousandtwohundredfiftyfive
  113     2000703552  twobillionsevenhundredthree|thousandfivehundredfiftytwo
  114     2000710285  twobillionsevenhundredtent|housandtwohundredeightyfive
  115     2000710582  twobillionsevenhundredtent|housandfivehundredeightytwo
  116     2000732355  twobillionsevenhundredthirtyt|wothousandthreehundredfiftyfive
  117     2000732553  twobillionsevenhundredthirtyt|wothousandfivehundredfiftythree
  118     2000733255  twobillionsevenhundredthirtyt|hreethousandtwohundredfiftyfive
  119     2000733552  twobillionsevenhundredthirtyt|hreethousandfivehundredfiftytwo
  120     2000817255  twobillioneighthundredsevente|enthousandtwohundredfiftyfive
  121     2000817552  twobillioneighthundredsevente|enthousandfivehundredfiftytwo
  122     2000879285  twobillioneighthundredseventyn|inethousandtwohundredeightyfive
  123     2000879582  twobillioneighthundredseventyn|inethousandfivehundredeightytwo
  124     3000007055  threebillionseven|thousandfiftyfive
  125     3000107155  threebilliononehundredseven|thousandonehundredfiftyfive
  126     3000107551  threebilliononehundredseven|thousandfivehundredfiftyone
  127     3000207255  threebilliontwohundredseven|thousandtwohundredfiftyfive
  128     3000207552  threebilliontwohundredseven|thousandfivehundredfiftytwo
  129     3000307355  threebillionthreehundredseven|thousandthreehundredfiftyfive
  130     3000307553  threebillionthreehundredseven|thousandfivehundredfiftythree
  131     3000407455  threebillionfourhundredseven|thousandfourhundredfiftyfive
  132     3000407554  threebillionfourhundredseven|thousandfivehundredfiftyfour
  133     3000507555  threebillionfivehundredseven|thousandfivehundredfiftyfive
  134     3000607556  threebillionsixhundredseven|thousandfivehundredfiftysix
  135     3000607655  threebillionsixhundredseven|thousandsixhundredfiftyfive
  136     3000698434  threebillionsixhundredninetyeigh|tthousandfourhundredthirtyfour
  ...
10883   997000399055  ninehundredninetysevenbillionthree|hundredninetyninethousandfiftyfive
10884   998000063494  ninehundredninetyeightbillionsixty|threethousandfourhundredninetyfour
10885   998000064394  ninehundredninetyeightbillionsixty|fourthousandthreehundredninetyfour
10886   998000064493  ninehundredninetyeightbillionsixty|fourthousandfourhundredninetythree
10887  1000000611455  onetrillionsixhundredeleven|thousandfourhundredfiftyfive
10888  1000000611554  onetrillionsixhundredeleven|thousandfivehundredfiftyfour
10889  1000000700155  onetrillionsevenhundredt|housandonehundredfiftyfive
10890  1000000700551  onetrillionsevenhundredt|housandfivehundredfiftyone

The above spelled-out American integer names have the property that when the letters are replaced with their alphabetic rank (a=1, b=2, etc.), the product of the ranks to the left of the pipe is equal to the product of the ranks to the right of the pipe. Six of the integers (indicated by ==) allow an alternate split as the letter "a" is shifted from one side of the pipe to the other side (which has no effect on the products).

The idea is based on A372222 by Éric Angelini and Jean-Marc Falcoz wherein the ranks of the left/right parts of the split are summed instead of multiplied. I feel that this allows far too many solutions for the sequence to be interesting.

I have a full extended list of the above.

Thursday, May 30, 2024

Stable primes

Subsequent to my Stable pandigital numbers blog, I will now share my calculation of the first 2486689 stable primes. Reference A373117 to understand what is meant here by "stable".

      1 2
      2 3
      3 5
      4 7
      5 11
      6 101
      7 113
      8 131
      9 151
     10 157
     11 179
     12 181
     13 191
     14 311
     15 313
     16 317
     17 353
     18 373
     19 383
     20 419
     21 421
    ...
2486669 999986513
2486670 999986719
2486671 999986927
2486672 999987323
2486673 999988133
2486674 999988547
2486675 999989149
2486676 999989519
2486677 999989981
2486678 999990841
2486679 999993641
2486680 999994081
2486681 999994613
2486682 999995261
2486683 999995629
2486684 999996071
2486685 999997249
2486686 999997457
2486687 999998059
2486688 999998683
2486689 999998801

The entire list, which is complete to 10^9, is here. Since there are 50847534 primes less than one billion, the stable primes in this range are 4.89% of that total.

Monday, May 27, 2024

Stable pandigital numbers

The recent OEIS A373117 addition had me wondering how many of the 3265920 pandigitals were "stable" (or "balanced" in Éric Angelini's article). I found 135914 such:

     1  1023469875
     2  1023487695
     3  1023495876
     4  1023497658
     5  1023569748
     6  1023578496
     7  1023579468
     8  1023584976
     9  1023587649
    10  1023596478
    11  1023649857
    12  1023685497
    13  1023746895
    14  1023748596
    15  1023749568
    16  1023764958
    17  1023765849
    18  1023845796
    19  1023847659
    20  1023865479
    21  1024368975
   ...
135894  9876253014
135895  9876305421
135896  9876314250
135897  9876321540
135898  9876324051
135899  9876325104
135900  9876340251
135901  9876350214
135902  9876351024
135903  9876403512
135904  9876405132
135905  9876405213
135906  9876412503
135907  9876431052
135908  9876432105
135909  9876502413
135910  9876503142
135911  9876510432
135912  9876513024
135913  9876520314
135914  9876521043

The entire list is here. The vertical blades in the above are the seventh digit in the first 21 terms and the fourth digit in the final 21 terms. There are eighteen different ways in which ten-digit integers might be stable. Here are the possible digit-multipliers, in pandigitals the frequency of their occurrence, and the [smallest index, largest index]:

(0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
(-1, 1, 2, 3, 4, 5, 6, 7, 8, 9)
(-1, 0, 1, 2, 3, 4, 5, 6, 7, 8)
(-2, -1, 1, 2, 3, 4, 5, 6, 7, 8)
(-2, -1, 0, 1, 2, 3, 4, 5, 6, 7)
(-3, -2, -1, 1, 2, 3, 4, 5, 6, 7)
(-3, -2, -1, 0, 1, 2, 3, 4, 5, 6)        =>  2010  [60132, 135914]
(-4, -3, -2, -1, 1, 2, 3, 4, 5, 6)       => 14892  [11501, 135847]
(-4, -3, -2, -1, 0, 1, 2, 3, 4, 5)       => 37712  [ 4507, 135595]
(-5, -4, -3, -2, -1, 1, 2, 3, 4, 5)      => 36873  [ 1194, 134669]
(-5, -4, -3, -2, -1, 0, 1, 2, 3, 4)      => 31768  [  345, 128927]
(-6, -5, -4, -3, -2, -1, 1, 2, 3, 4)     => 11393  [   45, 110619]
(-6, -5, -4, -3, -2, -1, 0, 1, 2, 3)     =>  1266  [    1,  60133]
(-7, -6, -5, -4, -3, -2, -1, 1, 2, 3)
(-7, -6, -5, -4, -3, -2, -1, 0, 1, 2)
(-8, -7, -6, -5, -4, -3, -2, -1, 1, 2)
(-8, -7, -6, -5, -4, -3, -2, -1, 0, 1)
(-9, -8, -7, -6, -5, -4, -3, -2, -1, 1)

Wednesday, May 22, 2024

Clockwise analog clock primes

Éric Angelini and Michael Branicky have added a version 2 of clockwise analog clock primes. I was sufficiently motivated thereby to extend Michael's a(59) [= a(28) of the original] having 1325 digits into the beyond.

Monday, April 22, 2024

Blossoms


It's the height of Toronto's cherry blossoms today, supposedly. It's also the start of our outside maple trees blossoming, which means soon enough they'll be littering the ground and it'll be impossible not to trek them into the house!

maple blossoms: April 25
maple blossoms give way to leaves: May 4
fallen maple blossoms: May 4
more fallen maple blossoms: May 6

Friday, April 19, 2024

Ed Pegg's product partition challenge

Now that Ed Pegg's recent Math-Fun suggestion is ensconced in the OEIS, I will highlight his assertion that the smallest product with a single-digit factorization is 1476395008. My idea is to enumerate a bunch of such integers by multiplying together all possible combinations of all possible powers of repdigits (of 2, 3, 4, 7, 8, 9), ignoring numbers larger than some limit. The products are then examined for having the nine digits that are not the factorization digit.

I managed to generate 2554 terms (<10^24) before running out of RAM. Michael Branicky upped this to 10000 terms (available as a b-file in OEIS A372106). Here is how things start:

 1       1476395008 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*22
 2     116508327936 = 4*4*4*4*4*4*4*4*4*444444
 3     505627938816 = 4*4*4*4*4444*444444
 4     640532803911 = 7*7*7*7*7*7*7*777777
 5    1207460451879 = 3*33*33*333*333*3333
 6    1429150367744 = 8*8*8*8*8*8*8*88*88*88
 7    1458956660623 = 7*77*77*77*77*77*77
 8    3292564845031 = 7*7777*7777*7777
 9    3820372951296 = 44*44*444*4444444
10    5056734498816 = 2*2*2*2*2*2*2*2*2*2*22222*222222
11    6784304541696 = 2*2*2*2*2*2*2*22*22*222*222*2222
12    8090702381056 = 4*4*4*4*4*4*44444*44444
13    9095331446784 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*22*222*222
14   10757095489536 = 2*2*2*2*2*2*2*2*2*2*2*22*22*22*222*2222
15   10973607685048 = 22222*22222*22222
16   13505488366293 = 7*7*77*77*77*777*777
17   14913065975808 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*222222
18   38203732951296 = 44*44*444*44444444
19   44859347140608 = 2*2*2*2*2*2*2*2*2*2*2*2*222*222*222222
20   50567390498816 = 2*2*2*2*2*2*2*2*2*2*22222*2222222
21   52612606387341 = 9*9*9*9*9*9*99*999999
22   76259892101481 = 3*3*3*3*3*3*3*3*3*3*33*33*33*33*33*33
23   88990517231616 = 4*4*4*4*4*44*4444*444444
24   89405043019776 = 2*2*2*2*22*22*22*22*22*22*222*222
25   97801459531776 = 2*2*2*2*2*2*2*2*2*2*2*2*22*22*222*222222
26  109737064485048 = 22222*22222*222222
27  119706531338304 = 222*222*222*222*222*222
28  124004938635963 = 7*7*7*77*777*777*7777
29  130043698937856 = 2*2*2*2*2*2*2*2*2*22*22*22*22*22*222*222
30  141759347490816 = 2*2*2*2*2*2*2*2*22*22*22*22*22*22*22*222
31  154530459877376 = 2*2*2*2*2*2*2*22*22*22*22*22*22*22*22*22
32  187619251060736 = 4*4*4*4*44*44*44*44*44*4444
33  191190753643648 = 2*2*2*22*22*2222*22222222

I had put the fully factored 10000 terms here but Neil saw fit to add it to the OEIS.

Saturday, April 13, 2024

Pandigital products

Based on a Neil Sloane misreading of an Ed Pegg idea (link has 187511 products, 10 MB):

 8596 = 2*14*307
 8790 = 2*3*1465
 9360 = 2*4*15*78
 9380 = 2*5*14*67
 9870 = 2*3*1645
10752 = 3*4*896
12780 = 4*5*639
14760 = 5*9*328
14820 = 5*39*76
15628 = 4*3907
15678 = 39*402
16038 = 27*594 = 54*297
16704 = 9*32*58
17082 = 3*5694
17820 = 36*495 = 45*396
17920 = 8*35*64
18720 = 4*5*936
19084 = 52*367
19240 = 8*37*65
20457 = 3*6819
20574 = 6*9*381
20754 = 3*6918
21658 = 7*3094
24056 = 8*31*97
24507 = 3*8169
25803 = 9*47*61
26180 = 4*7*935
26910 = 78*345
27504 = 3*9168
28156 = 4*7039
28651 = 7*4093
30296 = 7*8*541
30576 = 8*42*91
30752 = 4*8*961
31920 = 5*76*84
32760 = 8*45*91
32890 = 46*715
34902 = 6*5817
36508 = 4*9127
47320 = 8*65*91
58401 = 63*927
65128 = 7*9304
65821 = 7*9403

Neil has fast-tracked this into the OEIS.

Monday, April 08, 2024

A Falcoz digit-fancy

click to enlarge

In addition to my own "fanciful extension" of Éric Angelini's Two identical digits effort, Jean-Marc Falcoz suggested his own variation (at the end of the blog entry): "Lexicographically earliest sequence of distinct positive terms such that [the product of adjacent terms] contains exactly 1 digit 1 (if 1 is present), 2 digits 2 (if 2 is present), 3 digits 3 (if 3 is present), ... 9 digits 9 (if 9 is present)." He presented 113 terms of the sequence but I was hungry for more.

My initial plot (above) just exceeds 1000 terms. My updated (May 13) plot (below) extends this to 5000 terms. Term #4367 = 1785221551, a local maximum. Term #2213 = 2. Also known to appear are 8, 11, 13, 17, 22, 44, 84, 97, ... Possible products are A108571. Our indexed products are such that product #2 is term #2 multiplied by term #1 (product #1 is 1 by fiat). In the current product list we have 16 duplicates and 2 triplicates (#3 = #2215 = #2219 and #416 = #2349 = #3632). If typed by their constituent digits, regardless of digit order, the number of possible types is given by A125573. Our current list realizes 105 of these, the number of which (sorted by product digit-length) are: 1, 1, 1, 2, 1, 3, 4, 4, 6, 7, 8, 11, 10, 14, 13, 11, 7, 1.

click to enlarge

Thursday, March 28, 2024

Stymied by the beast

click to enlarge

The following described sequence is a somewhat fanciful extension of Éric Angelini's latest effort, Two identical digits. In my version, the products of adjacent terms must contain a single occurrence of the three digits "666". Furthermore, those products may not contain any additional sixes. The lexicographically earliest sequence of distinct positive terms starts:

1, 666, 10, 1666, 4, 1665, 40, 1667, 400, 4165, 16, 4163, 160, 4164, 1600, 4167, 64, 1041, 640, 1042, 373, 42, 1111, 15, 444, 150, 2444, 506, 361, 988, 675, 395, 422, 203, 1422, 469, 398, 268, 1368, 173, 385, 329, 154, 303, 55, 1211, 406, 211, 79, 654, 316, 1635, 579, 616, 1082, 191, 349, 234, 285, 269, 114, 585, 849, 903, 738, 257, 2166, 251, 166, 751, 355, 723, 142, 223, 299, 534, 312, 1047, 78, 47, 780, 470, 1078, 547, 195, 188, 461, 906, 961, 694, 96, 2778, 6, 111, 60, 611, 551, 121, 146, 621, 365, 484, 427, 156, 235, 539, 94, 39, 171, 390, 940, 922, 453, 1273, 288, 926, 18, 37, 180, 370, 991, 148, 45, 1037, 450, 1480, 518, 287, 929, 61, 306, 561, 101, 165, 1010, 264, 1351, 493, 338, 286, 233, 402, 733, 773, 345, 483, 138, 157, 552, 1407, 474, 109, 1896, 879, 292, 605, 573, 242, 73, 913, 730, 1242, 537, 869, 767, 598, 267, 624, 1068, 1186, 181, 186, 1181, 302, 383, 174, 159, 419, 214, 1246, 271, 2458, 556, 1198, 367, 198, 867, 769, 997, 669, 548, 1034, 49, 34, 196, 85, 549, 340, 490, 136, 1225, 544, 1226, 87, 318, 524, 757, 273, 244, 683, 449, 438, 207, 322, 353, 1237, 1024, 1627, 84, 793, 248, 1075, 62, 43, 155, 172, 562, 1542, 123, 2542, 262, 636, 435, 613, 882, 113, 59, 452, 590, 774, 559, 477, 58, 1149, 145, 1839, 294, 339, 1294, 515, 712, 936, 178, 206, 411, 892, 747, 278, 959, 374, 459, 363, 382, 541, 308, 671, 993, 1678, 588, 1133, 802, 532, 1441, 226, 118, 387, 689, 194, 756, 1940, 1189, 485, 1374, 558, 727, 243, 686, 431, 1547, 237, 218, 948, 545, 1223, 1003, 222, 3, 1222, 30, 2220, 300, 2221, 546, 122, 153, 305, 612, 1939, 1071, 249, 589, 283, 702, 95, 807, 38, 307, 152, 1557, 538, 57, 117, 570, 1169, 714, 169, 572, 641, 182, 163, 1182, 663, 553, 522, 53, 1257, 530, 1258, 177, 258, 677, 645, 708, 758, 927, 719, 51, 366, 510, 915, 204, 866, 77, 606, 44, 2197, 440, 3106, 261, 106, 629, 265, 1044, 638, 407, 1466, 201, 466, 143, 662, 643, 648, 1028, 843, 791, 126, 529, 315, 1291, 284, 446, 71, 1446, 571, 467, 798, 835, 499, 1837, 818, 337, 999, 334, 1499, 378, 388, 1718, 1487, 603, 608, 1261, 1060, 761, 219, 414, 161, 706, 1161, 574, 259, 296, 225, 1185, 436, 1528, 1863, 358, 27, 247, 270, 2467, 404, 66, 601, 610, 765, 871, 421, 346, 684, 536, 199, 737, 904, 295, 565, 472, 1137, 586, 455, 652, 1022, 212, 739, 902, 1483, 989, 394, 489, 1362, 372, 905, 593, 281, 344, 775, 86, 31, 215, 124, 457, 938, 711, 375, 1511, 441, 582, 252, 2209, 845, 789, 376, 975, 1094, 1432, 277, 858, 439, 653, 1021, 555, 12, 1389, 48, 1388, 480, 3471, 46, 471, 184, 362, 93, 717, 298, 1717, 33, 505, 330, 808, 1283, 1377, 397, 445, 824, 2022, 577, 658, 770, 865, 771, 246, 1271, 139, 1494, 473, 141, 26, 1141, 260, 1410, 1026, 65, 564, 526, 391, 2526, 891, 187, 918, 1053, 633, 911, 622, 403, 488, 1365, 884, 1727, 965, 691, 82, 313, 820, 1313, 1117, 597, 1116, 454, 779, 599, 734, 99, 673, 990, 1683, 102, 183, 255, 732, 91, 293, 876, 1035, 644, 647, 103, 356, 468, 698, 617, 886, 231, 202, 132, 2020, 533, 125, 1333, 2, 333, 20, 833, 8, 2083, 80, 3333, 5, 1332, 50, 3332, 200, 1833, 511, 424, 393, 1391, 479, 254, 656, 2540, 853, 781, 854, 379, 1073, 2242, 1234, 443, 462, 772, 1019, 193, 962, 1193, 475, 1403, 354, 129, 517, 98, 17, 392, 170, 980, 68, 245, 272, 2449, 634, 736, 2264, 692, 342, 423, 1316, 1654, 790, 1097, 945, 705, 52, 1281, 520, 1859, 974, 359, 1856, 1329, 923, 939, 497, 1341, 609, 602, 1107, 838, 107, 2492, 729, 503, 888, 75, 889, 750, 2222, 21, ...

The region from 740000 to 890000 is detailed here:

click to enlarge