Skip to main content

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.

Labels:

Comments

Post a Comment

Popular posts from this blog

The Number 1.6: Base of Fibonacci Sequence and the Golden Ratio

This post is about the number 1.6 that forms the base of the mathematical concepts Fibonacci sequence and the Golden ratio. For better understanding of this amazing number 1.6, it is important to know about Fibonacci sequence and the Golden ratio. Golden Ratio The  Golden ratio is the ratio of a line segment cut into two pieces of distinct lengths such that the ratio of the longer segment to the shorter segment is equal to the whole segment to that of the longer segment. It is denoted by the symbol Phi [ø]. Suppose there is a line segment AB, and C is any point that divides the line segment AB into two pieces of distinct lengths such that AC is greater than CB. Then, as per the definition, [ø] = AC/CB = AB/AC. The value of the Golden ratio [ø] is 1.6 1803398875 . As you can see, the value of the Golden ratio starts with the number 1.6. For more information on the Golden ratio, see https://theamazingconcepts.blogspot.com/2021/07/the-golden-ratio-key-to-amazing-design....
Post a Comment
Read more

Kaprekar Constant 6174 for Four-digit Number

Dattatreya Ramchandra Kaprekar (D. R. Kaprekar; 1905 – 1986) was an Indian mathematician who discovered several classes of natural numbers including Kaprekar Constant. The number 6174 is called Kaprekar Constant for four-digit number. Why is the number 6174 called Kaprekar Constant? In 1949, D. R. Kaprekar discovered an interesting property of the number 6174 through the mathematical calculations. D. R. Kaprekar observed that when any four-digit number of non-identical digits (not all digits are same) is arranged in two parts – highest number and lowest number, and difference is calculated using subtraction method, then the end-result will be always 6174 either in the first step or in no more than 7 steps. If the process is continued even after getting the end-result 6174, then the end-result will be always 6174 at each Iteration, therefore the number 6174 is known as Kaprekar Constant. Steps to Follow Take any four-digit number with different digits. Form the highest nu...
Post a Comment
Read more

Universe Secret Code 369 Theory

When it comes to the origin of the universe secret code 369 theory, one name associated with this amazing concept is ‘Nikola Tesla’. Nikola Tesla [10 July 1856 – 07 January 1943] was not only a great inventor / scientist, but also the greatest mind of all the time. He has done a plenty of research throughout his life, and highly recognized for his contributions towards the following : High Voltage, High Frequency Power Experiments Alternating Current (AC) Electricity Supply System Apart from the above-mentioned contributions, he has given ‘Universe Secret Code 369 Theory’ and has called the number 369 – the key to the universe. Now the question that will come to the curious minds is that why Nikola Tesla has called this number the key to the universe. The answer to this question is described in this post. Let start with a specific pattern as shown below: 1 1 + 1 => 2 2 + 2 => 4 4 + 4 => 8 8 + 8 => 16; 1 + 6 => 7 16 + 16 => 32; 3 + 2 => 5 >>>>>>...
Post a Comment
Read more