A number that can be expressed as
For example, 648 is the first ABA number with two representations,
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, for an odd example,
In general, it is possible to construct a number with an arbitrary number
of representations, of the form
are the prime numbers 2, 3, 5,.... and clearly
, it is possible to find the
's, and thus
, by solving the system of
is the product of the primes from
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
Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
A graph displaying how many ABA numbers are multiples of the primes p
from 2 to 71. In black the ideal line 1/p
. Among the first 2252741 terms, 2245081 are even.