Primesalso

Primesalso is a term used in recreational number theory to describe a simple relation to prime numbers. It is not a standard mathematical notion, and different sources may define it differently. The most common definition states that a natural number n is primesalso if either n is prime or n−1 is prime. In effect, primesalso consists of all primes together with all numbers that lie immediately to the right of a prime.

Examples include 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 17, 18, 19, 20, 23, 24, 29, 30, 31, 32. The number 9 is not primesalso, since 9 is not prime and 9−1=8 is not prime; similarly 10 is not included, as neither 10 nor 9 is prime.

Properties and notes: The set has natural density zero, since primes up to x are about x/log x and primesalso up to x are about 2x/log x. The primesalso elements thus appear relatively sparsely among the integers, with a tendency to cluster around primes and their successors. The construction interacts simply with primes: every prime is included, and for each prime p the number p+1 is included as well (except for overlaps like 3, which is both prime and a successor of 2).

Variants: Some casual writings define primesalso as numbers that are prime or one less than a prime (i.e., n is prime or n+1 is prime), which yields a slightly different sequence and may include 1. In formal contexts, primesalso is not an established term and definitions may vary.

See also: prime numbers, prime gaps, twin primes, almost primes.