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):
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).
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