## Saturday, June 29, 2024

### 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.

## 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 Ã‰ric 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.