Practical Number

In number theory, a Practical number or Panarithmic number is a positive integer n that can represent all smaller numbers m (m < n) as the sums of distinct divisors of n.

In other words, any positive integer n having the property that all smaller integers can be represented as the sums of distinct divisors of integer n is referred to as a Practical number or Panarithmic number.

The Procedure

Step 1: Consider a positive integer n.

Step 2: Find out the distinct divisors of n.

Step 3: Check whether all numbers smaller than n can be represented as the sums of distinct divisors of n.

Step 4: If all numbers smaller than n can be represented as the sums of distinct divisors of n, then n is a Practical number or Panarithmic number, else not.

Practical Number Sequence

1, 2, 4, 6, 8, 12, 16, 18, 20, ….., n.

For Example

Integer: 24

Distinct divisors of 24: 1, 2, 3, 4, 6, 8, 12.

All smaller numbers such as:

5 => 2 + 3

7 => 3 + 4

9 => 3 + 6

10 => 4 + 6

11 => 3 + 8

13 => 1 + 12

14 => 2 + 12

15 => 3 + 12

16 => 4 + 12

17 => 1 + 4 + 12

18 => 6 + 12

19 => 1 + 6 + 12

20 => 8 + 12

21 => 1 + 8 + 12

22 => 2 + 8 + 12

23 => 3 + 8 + 12

can be represented as the sums of distinct divisors of integer 24. Hence integer 24 is a Practical number or Panarithmic number.

Hope you have understood this amazing concept of Practical number or Panarithmic number. In case of any issue, provide your valuable comments.