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.

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