Critters

I wasn't happy when I saw this rat in our back yard last Tuesday. I know, rats have had bad press over the centuries. Nevertheless, its presence indicates to me that my attempts at keeping critters on the other side of the fence (climbers like squirrels and raccoons obviously exempted) have once again failed. A quick perimeter check found a hole under the west fence, halfway down the extended driveway, but my prejudice is that this was skunk dug. I expect the rats to come from under the back-of-the-property shed and/or the east fence (where I know Bodie smells them just on the other side). Anyways, I filled the passage with dirt, placed a small piece of concrete on top, and hoped for the best.

It isn't all bad news. This morning I spotted a chipmunk (a neighbourhood rarity) on the deck:

Next-day update: Bodie flushed a rat out from the hosta/bushes thicket in the backyard and I saw it run to the east fence where I had a large, heavy patio stone leaning against a leaning fence post (hoping to counterforce its lean). Moving the stone, I could see that the area at the base of the post had been compromised and that this was the conduit by which the rat was able to breach the fence. I stuffed the hole with rocks and a brick (to hold the rocks in place). I will still need to be vigilant over the course of the summer.

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).

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. The sequence is now in the OEIS. I have also added a large-number examples file evaluated at 10^3n up to 10^303.