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A cluster prime is a prime p such that every even natural number k ≤ p − 3 is the ... Primes p that divide 2 n − 1, for some prime number n. 3, 7, 23, 31, 47 ...
A number where some but not all prime factors have multiplicity above 1 is neither square-free nor squareful. 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.
This is due to the Euclid–Euler theorem, partially proved by Euclid and completed by Leonhard Euler: even numbers are perfect if and only if they can be expressed in the form 2 p−1 × (2 p − 1), where 2 p − 1 is a Mersenne prime. In other words, all numbers that fit that expression are perfect, while all even perfect numbers fit that form.
Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are even, and decimal numbers that end in 0 or 5 are divisible by 5. [11]
Plot of the number of divisors of integers from 1 to 1000. Highly composite numbers are in bold and superior highly composite numbers are starred. ... 100: 1, 2, 4, 5 ...
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.
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, ...
n even with an odd number of prime factors 1 2 9 21 100 180 1010 6067 + + n even with an odd number of distinct prime factors 3 4 21 49 268 482 2486 4452 + + n odd with an odd number of prime factors 3 4 23 43 251 437 2428 4315 + + n odd with an odd number of distinct prime factors 4 5 28 56 317 566 3070 5607 + + n even squarefree with an even ...