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tau or refactorable numbers
R.E.Kennedy and C.N.Cooper called    a tau number if    is divisible by the number    of its divisors, and they proved that the natural density of tau numbers is zero.

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

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