Mixedprime

Mixedprime is a term used in some recreational number theory contexts to denote a natural number that blends the properties of primes and semiprimes. There is no universally accepted definition, and the term is not standard in formal mathematics. A common informal definition treats a mixedprime as a natural number that is either a prime or a semiprime, where a semiprime is the product of two primes (counting multiplicity).

Under this definition, the mixedprimes consist of all primes together with semiprimes such as 4 (2×2), 6 (2×3), 9 (3×3), 10 (2×5), 14 (2×7), 15 (3×5), 21 (3×7), and 22 (2×11). If one requires the two primes to be distinct, the numbers are the squarefree semiprimes (for example, 6, 10, 14, 15, 21, 22) while excluding 4 and 9.

Properties include that the set of mixedprimes is infinite. Relative to the positive integers, their density tends to zero. The counting function S(x), the number of mixedprimes not exceeding x, is asymptotically similar to the semiprime count and satisfies S(x) ~ x log log x / log x as x grows large.

Computation and analysis: To generate mixedprimes up to N, one can list all primes up to N and form products p×q ≤ N, or simply append the primes themselves. In cryptography and number theory, semiprimes are studied for factoring difficulty; the mixedprime concept highlights both primes and semiprimes.

History and usage: The term mixedprime is informal and appears in some puzzle references or glossary-style articles; formal literature typically uses "prime" and "semiprime" as distinct classes. Related concepts include primes, semiprimes, and almost primes.