## Friday, 1 March 2013

### FRIDAY, 1 MARCH 2013

Today is the $60^{th}$ day of the year.

$60 = 2^2 \times 3 \times 5$

$60 = 29 + 31 = 11 + 13 + 17 + 19$ that is $60$ can be expressed as the sum of consecutive primes in two ways.

$60$ is a member of the following Pythagorean triples:
$(11, 60, 61)$
$(60, 91, 109)$
$(60, 221, 229)$
$(60, 899, 901)$

$60$ is the twelfth number in Narayana's Cows Sequence, see A000930 and here.

A natural number $u$ is a unitary divisor of a natural number $n$ if, and only if, $u$ divides $n$ and $\frac {n}{u}$ has no factors in common with $u$, that is $\frac {n}{u}$ is coprime with $u$.
For example $15$ is a unitary divisor of $60$ because $\frac {60}{15} = 4$ and $4$ is coprime with $15$.
Whereas $\frac {60}{30} = 2$ and $2$ is not coprime with $30$.
Thus the unitary divisors of $60$ are $1, 3, 4, 5, 12, 15$ and $20$.
Now, the sum of the unitary divisors of $60$ is $60$ i.e. $1 + 3 + 4 + 5 + 12 + 15 + 20 = 60$ this means that $60$ is known as a Unitary Perfect Number, see A002827 and here.
The first five Unitary Perfect Numbers are $6,60,90,87360,146361946186458562560000$. The sixth Unitary Number has yet to be found. This paper postulates that the sixth Unitary Perfect Number has to be larger than 24 digits.