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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)