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The Liouville function λ(n) is 1 if Ω(n) is even, and is -1 if Ω(n) is odd. The Möbius function μ(n) is 0 if n is not square-free. Otherwise μ(n) is 1 if Ω(n) is even, and is −1 if Ω(n) is odd. A sphenic number has Ω(n) = 3 and is square-free (so it is the product of 3 distinct
Even and odd numbers: An integer is even if it is a multiple of 2, and is odd otherwise. Prime number: A positive integer with exactly two positive divisors: itself and 1. The primes form an infinite sequence 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...
The number is taken to be 'odd' or 'even' according to whether its numerator is odd or even. Then the formula for the map is exactly the same as when the domain is the integers: an 'even' such rational is divided by 2; an 'odd' such rational is multiplied by 3 and then 1 is added.
a primitive abundant number is an abundant number whose proper divisors are all deficient numbers or they are "abundant" numbers with no abundant proper divisors; a weird number is an abundant number that is not semiperfect; that is, no subset of the proper divisors of n sum to n
The even and odd numbers alternate. Starting at any even number, counting up or down by twos reaches the other even numbers, and there is no reason to skip over zero. [8] With the introduction of multiplication, parity can be approached in a more formal way using arithmetic expressions. Every integer is either of the form (2 × ) + 0 or (2 × ...
120 is . the factorial of 5, i.e., ! =.; the fifteenth triangular number, [2] as well as the sum of the first eight triangular numbers, making it also a tetrahedral number. 120 is the smallest number to appear six times in Pascal's triangle (as all triangular and tetragonal numbers appear in it).
A cluster prime is a prime p such that every even natural number k ≤ p − 3 is the difference of two primes not exceeding p. 3, 5, 7, 11, 13, 17, 19, 23, ... (OEIS: A038134) All odd primes between 3 and 89, inclusive, are cluster primes. The first 10 primes that are not cluster primes are: 2, 97, 127, 149, 191, 211, 223, 227, 229, 251.
Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2 p − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1.