Later, S.Colton, by means of his automatic concept formation program HR, re-discovered them and called these numbers refactorable.
S.Colton proved that there are infinite tau numbers, since for every prime
the number
is refactorable and that
all the odd tau numbers are squares and that tau numbers
are congruent to 0, 1, 2 or 4
.
Moreover, Colton proved that, if is the product of
distinct primes, then there are
tau numbers with
divisors.
For example, the tau numbers with
divisors are 720, 1200, 1620, 4050, 7500 and 11250.
It is easy to see that, if and
and
are
tau numbers, then
is also a tau number.
J.Zelinsky proved in 2002 that there are no 3 consecutive tau numbers. However, it is conjectured that there are infinite pairs of consecutive tau numbers, which, as Colton proved, always contain a square. The first ones are (1, 2), (8, 9), (1520, 1521) and (50624, 50625).
Below, the spiral pattern of tau numbers up to . See the page on prime numbers for an explanation and links to similar pictures.
The smallest Pythagorean triple of tau numbers is (40, 96, 104). Is there a primitive such triple?
The first tau numbers are 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225 more terms