Mathematics UIL: Types of Numbers
extravagant or wasteful
a natural number that has less digits than its prime factorization (including exponents) ex: 4, 6, 8, 9, 12, 18, 20, 22
frugal or economical
a natural number that has more digits than its prime factorization (including exponents) ex: 125 (5³), 128 (2⁷), 243 (3⁵)
equidigital
a natural number that has the same number of digits as its prime factorization (including exponents) ex: 1, 2, 3, 5, 7, 10, 11, 13, 14, 15
primeval
a natural number where using some or all of its digits to make prime numbers will give you more prime numbers than a previous number ex: 2, 13, 37, 107, 113
perfect
a number in which the sum of its proper divisors is equal to the number, the first four being 6, 28, 496, 8128 ex: 6, 28
abundant
a number in which the sum of its proper divisors is greater than the number, the first few of which are 12, 18, 20, 24 ex: 12, 18, 20, 24, 30, 36, 40, 42
deficient
a number in which the sum of its proper divisors is less than the number ex: 1, 2, 3, 4, 5, 7, 8, 9, 10
transcendental
a number that is not a root of a non-zero polynomial equation with rational coefficients ex: π and e
evil
a number which has an even number of ones in its binary expansion ex: 3 (11₂), 5 (101₂), 6 (110₂), 9 (1001₂), 10 (1010₂)
odious
a number which has an odd number of ones in its binary expansion ex: 1 (1₂), 2 (10₂), 4 (100₂), 7 (111₂), 8 (1000₂)
natural number
1, 2, 3... (counting numbers)
integer number
-2, -1, 0, 1, 2...
whole number
0, 1, 2, 3...
Which numbers are obviously abundant?
any multiple of a perfect number (6, 28)
Which numbers are obviously deficient?
any power of a prime
polite
can be written as the sum of two or more consecutive integers ex: 3, 5, 6, 7, 9, 10, 11 (all numbers except for powers of 2 are polite)
impolite
can't be written as the sum of two or more consecutive integers ex: 1, 2, 4, 8, 16, 32, 64 (powers of 2 are the only impolite numbers)
Which numbers deal with the sum of a number's proper divisors?
deficient, perfect, and abundant
Which numbers deal with the digits in prime factorization?
frugal or economical, equidigital, and extravagant or wasteful
Which numbers deal with the continous sum of the squares of a number's digits?
happy and unhappy
integral value
integer
lucas
like the Fibonacci sequence, this sequences has a number that is the sum of two previous numbers in the sequence; however, this sequence involves 2, 1, 3, 4, 7...; Fibonacci's involves 1, 1, 2, 3, 5... ex: 2, 1, 3, 4, 7, 11, 18, 29
Which numbers deal with eliminating every other number, then every third number, etc.?
lucky and unlucky
unlucky
not a lucky number ex: 2, 4, 5, 6, 8, 10, 11, 12, 14
Which numbers deal with the number of ones in a number's binary expansion?
odious and evil
Which numbers deal with being the sum of consecutive integers?
polite and impolite
happy
replace a number with the sum of the squares of its digits and continue this process; for this set, the number 1 will be reached ex: 1, 7, 10, 13, 19, 23, 28, 31, 32, 44
unhappy
replace a number with the sum of the squares of the digits and continue this process; for this set, an endless loop which will not end with 1 will be created ex: 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15
lucky
take the natural numbers; eliminate the even numbers; the next number is 3, so eliminate all third numbers from the remaining list; the next number is 7, so eliminate all seventh numbers from the list; continue this process; all remaining numbers are part of this set ex: 1, 3, 7, 9, 13, 15, 21
