19: Ten Special Types of Numbers in .NET framework

Create QR Code 2d barcode in .NET framework 19: Ten Special Types of Numbers

A perfect number is a number in which the sum of its proper divisors (numbers smaller than the number in question that divide it evenly) is equal to the number in question For instance, the number 6 is a perfect number, because 1 + 2 + 3 = 6 The next few perfect numbers are 28 (1 + 2 + 4 + 7 + 14), 496 (1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248), and 8,128 Notice how these proper divisors all have pairs, except for the 1 The pairs multiply together to give you the number in question: 2 times 14, 4 times 7 in the case of the 28 Can you find all the proper divisors of 8,128 Have fun! There s no magic formula for finding these numbers Just get out your calculator!

Two numbers are amicable if each value is equal to the sum of the proper divisors of the other The numbers 284 and 220 are amicable, for example, because the sum of the proper divisors of 284 is 220 (1 + 2 + 4 + 71 + 142), and the sum of the proper divisors of 220 is 284 (1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110) Other sets of amicable numbers include 1,184-1,210 and 17,296-18,416 Do you want to find more amicable numbers Go for it, but don t expect a magic formula to help you

A number is considered happy if the sum of the squares of its digits, or the sum of the squares of the sum of its squares, is equal to one For instance, the number 203 is happy because the sum of the squares of its digits is 22 + 02 + 32 = 4 + 9 = 13; the sum of the squares of the digits in 13 is 12 + 32 = 1 + 9 = 10; and the sum of the squares of the digits in 10 is 12 + 02 = 1 Finally! You may be surprised to find that a bigger number such as 2,211 is happy in fewer steps! You take 22 + 22 + 12 + 12 = 4 + 4 + 1 + 1 = 10, and 12 + 02 = 1

An abundant number is a number with one rule: The sum of its divisors must be greater than twice the number For instance, the number 12 is abundant because 1 + 2 + 3 + 4 + 6 + 12 = 28, and 28 is greater than twice 12 Some other abundant numbers include 18, 20, 24, 30, 36, and 100

A deficient number has almost the opposite rule of an abundant number (see the previous section) If the sum of the proper divisors of a number is less than the original number, the number is a deficient number An example of a deficient number is 15, because the sum of its proper divisors is 1 + 3 + 5 = 9, and 9 is less than 15 What about the number 32 Is it deficient The sum of its proper divisors is 1 + 2 + 4 + 8 + 16 = 31, so 32 is also deficient

A narcissistic number is a number that you can write by using operations involving all the digits in the number For instance, the number 371 is narcissistic because if you take the digits 3, 7 and 1 raise each to the third power, and add the powers together, you get the original number: 33 + 73 + 13 = 27 + 343 + 1 = 371 The digits of the number look in the mirror with operations, so to speak, and see only themselves Another narcissistic number is 2,427, because you can raise the digits to successively larger powers, add up the powers, and get the original number: 21 + 42 + 23 + 74 = 2 + 16 + 8 + 2,401 = 2,427 The possibilities for self-absorbed numbers are endless