Narcissistic Numbers, Weird Numbers, and Fortunate PrimesDate: 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/ |
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