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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, ... Composite number: A positive integer that can be factored into a product of smaller positive integers. Every integer greater than one is either prime or composite.
A positive integer with more divisors than any smaller positive integer. A002182: Superior highly composite numbers: 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, ... A positive integer n for which there is an e > 0 such that d(n) / n e ≥ d(k) / k e for all k > 1. A002201: Pronic numbers: 0, 2, 6, 12, 20, 30, 42, 56, 72 ...
Informally, a sequence has a limit if the elements of the sequence become closer and closer to some value (called the limit of the sequence), and they become and remain arbitrarily close to , meaning that given a real number greater than zero, all but a finite number of the elements of the sequence have a distance from less than .
The first ordinal number that is not a natural number is expressed as ω; this is also the ordinal number of the set of natural numbers itself. The least ordinal of cardinality ℵ 0 (that is, the initial ordinal of ℵ 0) is ω but many well-ordered sets with cardinal number ℵ 0 have an ordinal number greater than ω.
The limit of a sequence of powers of a number greater than one diverges; in other words, the sequence grows without bound: b n → ∞ as n → ∞ when b > 1. This can be read as "b to the power of n tends to +∞ as n tends to infinity when b is greater than one". Powers of a number with absolute value less than one tend to zero: b n → 0 as ...
An integer is positive if it is greater than zero, and negative if it is less than zero. Zero is defined as neither negative nor positive. The ordering of integers is compatible with the algebraic operations in the following way: If a < b and c < d, then a + c < b + d; If a < b and 0 < c, then ac < bc
An abundant number whose abundance is greater than any lower number is called a highly abundant number, and one whose relative abundance (i.e. s(n)/n ) is greater than any lower number is called a superabundant number; Every integer greater than 20161 can be written as the sum of two abundant numbers. The largest even number that is not the sum ...
Unlike n-smooth numbers, for any positive integer n there are only finitely many n-powersmooth numbers, in fact, the n-powersmooth numbers are exactly the positive divisors of “the least common multiple of 1, 2, 3, …, n” (sequence A003418 in the OEIS), e.g. the 9-powersmooth numbers (also the 10-powersmooth numbers) are exactly the ...