## 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).