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[6] [7] When x {\displaystyle x} is a positive integer, ( x ) n {\displaystyle (x)_{n}} gives the number of n -permutations (sequences of distinct elements) from an x -element set, or equivalently the number of injective functions from a set of size n {\displaystyle n} to a set of size x {\displaystyle x} .
Ω(n), the prime omega function, is the number of prime factors of n counted with multiplicity (so it is the sum of all prime factor multiplicities). A prime number has Ω( n ) = 1. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 (sequence A000040 in the OEIS ).
If none of its prime factors are repeated, it is called squarefree. (All prime numbers and 1 are squarefree.) For example, 72 = 2 3 × 3 2, all the prime factors are repeated, so 72 is a powerful number. 42 = 2 × 3 × 7, none of the prime factors are repeated, so 42 is squarefree. Euler diagram of numbers under 100:
Name First elements Short description OEIS Mersenne prime exponents : 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, ... Primes p such that 2 p − 1 is prime.: A000043 ...
In number theory, the prime omega functions and () count the number of prime factors of a natural number . Thereby (little omega) counts each distinct prime factor, whereas the related function () (big omega) counts the total number of prime factors of , honoring their multiplicity (see arithmetic function).
Here the exponent () is the multiplicity of as a prime factor of (also known as the p-adic valuation of ). For example, in base 10, 378 = 2 1 · 3 3 · 7 1 is a Smith number since 3 + 7 + 8 = 2 · 1 + 3 · 3 + 7 · 1, and 22 = 2 1 · 11 1 is a Smith number, because 2 + 2 = 2 · 1 + (1 + 1) · 1.
Note that our covenant maximum will fall throughout the year, decreasing to 5.25 times in the second quarter, five times in the third quarter and 4.75 times in the fourth quarter of this fiscal year.
(The place value is the factorial of one less than the radix position, which is why the equation begins with 5! for a 6-digit factoradic number.) General properties of mixed radix number systems also apply to the factorial number system.