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ABA numbers
A number that can be expressed as  , for  .

For example, 648 is the first ABA number with two representations,    and  .

According to Robert Munafo, the smallest number with 3 such representations is 344373768, which is equal to  ,    and  .

It is not difficult to build larger examples of numbers with 3 representations, but they actually tend to be quite larger. For example,

or
or, for an odd example,

In general, it is possible to construct a number    with an arbitrary number of representations, of the form

where the    are the prime numbers 2, 3, 5,.... and clearly  . Expressing    as  , it is possible to find the exponents  's, and thus  , by solving the system of    modular equations
where    is the product of the primes from    to    and    ranges from    to  .

For example, a number of 255 digits with 4 such representations is    and one of 2549 digits with 5 representations is  .

The smallest Pythagorean triples made of ABA numbers are (98304, 131072, 163840) and (229376, 786432, 819200) which correspond to    and  .

The first ABA numbers are 8, 18, 24, 32, 50, 64, 72, 81, 98, 128, 160, 162, 192, 200 more terms

ABA numbers can also be... (you may click on names or numbers and on + to get more values)