Narcissistic Numbers, Weird Numbers, and Fortunate Primes

Date: 03/27/98 at 06:19:00
From: Kenny Tang
Subject: meanings of words

Dear Dr. Maths, 

I've tried many times to find a site to find the meanings of:

Narcissistic Numbers 
Weird Numbers
Fortunate Numbers

Can you tell me any sites that define these terms, or can you give me 
the meanings of these terms? Also, can you give me examples so I could 
understand more of what the words mean?

Many thanks for your co-operation and help!

Your sincerely,
Kenny Tang

Date: 03/27/98 at 07:02:51
From: Doctor Allan
Subject: Re: meanings of words

Hi Kenny,

NARCISSISTIC NUMBERS:

DEFINITION
A narcissistic number is an n-digit number that is the sum of the 
n-th powers of its digits.

Examples:

     153 = 1^3 + 5^3 + 3^3.
     548834 = 5^6 + 4^6 + 8^6 + 8^6 + 3^6 + 4^6.

WEIRD NUMBER:

This requires several intermediary definitions.

DEFINITION
A perfect number is a number that equals the sum of its proper 
divisors (i.e., the divisors not including the number itself).

Example: 28 is a perfect number because 28 = 1 + 2 + 4 + 7 + 14.

DEFINITION
If an integer n is not a perfect number, we can look at the sum of 
proper diviors anyway, and we denote this by s(n).

DEFINITION
If s(n) > n, then the integer n is called an abundant number.

Example: 12 is an abundant number because

     s(12) = 1 + 2 + 3 + 4 + 6 = 16 > 12.

DEFINITION
A number that is the sum of some or all of its proper divisors is 
called a semiperfect number.

Example: 20 is a semiperfect number because 20 = 1 + 4 + 5 + 10.

DEFINITION
A weird number is then defined as a number that is abundant without 
being semiperfect.

Examples: 70 and 836 are weird numbers.

FORTUNATE PRIMES:

DEFINITION
                     k   
Define X_k := 1 + Product {p_i}
                    i=1

where the product is taken over primes (the p_i).

DEFINITION
Let q_k be the smallest prime that is greater than X_k.

CONJECTURE
R.F. Fortune conjectured that q_k - X_k + 1 is prime for all k, and 
these are called fortunate primes.

Example:

     Let k = 2. Then X_2 = 1 + (2*3) = 7.
     Therefore q_2 = 11.

     q_2 - X_2 + 1 = 11 - 7 + 1 = 5, which is prime.

I hope this was what you were looking for.

Sincerely,

-Doctor Allan, The Math Forum
Check out our web site! http://mathforum.org/dr.math/