Practical Numbers
\[n\]
is one for which every positive integer \[1 \le r \le n\]
can be written as a sum of positive divisors of \[n\]
. Example: The divisors of 6 ate 1, 2, 3, and 6.1=1
2=2
3=3
4=1+3
5=2+3
6=1+2+3=6
Hence 6 is a practical number.
Example: The divisors of 5 are 1, and 5.
1=1
2=2 but two is not a divisor of 5.
Example: The divisors of
\[2^n\]
are \[1, \; 2, \; 4,..., \; 2^n\]
.Any number
\[a_1\]
less or equal to than \[2^n\]
can be treated in the following way. Divide by the largest \[2^{j_1}\]
less than \[a_1\]
to get a remainder \[a_2\]
and divide the reminder by the largest power of \[2^{j_2}\]
less than \[a_2\]
. Continuing in this way, we eventually have a remainder of 0 or 1, and since 1 is a divisor of \[2^n\]
we can write \[2^n\]
as a sum of all those powers of 2 we divided by, plus either 0 or 1.Hence
\[2^n\]
is a practical number.
