mekso variable-arity (at most ternary) operator: number of prime divisors of number X_{1}, counting with or without multiplicity according to the value X_{2} (1 xor 0 respectively; see note for equality to -1 and for default value), in structure X_{3}.

X_{1} may be a number in a generalized sense: anything living in a ring with primes; most commonly, it will be a positive integer. Units are not considered to be prime factors for the purposes of this counting. x_{2} toggles the type of counting and must be exactly one element of Set(-1, 0, 1). If x_{2 }= -1 and X_{3} is the typical ring of integers (with the ordering here being the traditional ordering of the integers), then the output is k = sup(Set(i: i is a positive integer, and v_{p}, where: p_{i} is the ith prime (such that p_{1 }= 2), and v_{p} is the p-adic valuation (see: "pau'au") of the input; in other words, this mode yields the index i of the greatest prime p_{i} which has a nonzero power r_{i} such that p_{i^r} divides X_{1}; if X_{1} is a unit and x_{2 }= -1, then this word outputs 0; if X_{1 }= 0 and X_{2 }= -1, then this word outputs positive infinity; this mode counts early primes which have power 0 in the prime factorization of X_{1} but does not count the infinitely many later ones which occur after the last nonzero prime power in that factorization (when X_{1} is not 0 and is not a unit). If X_{2 }= 0, then the prime factors with nonzero power are counted without multiplicity (they are counted only uniquely and according to their distinctness, ignoring their exponents unless such is 0 (in which case, it is not counted)); in other words, under this condition, this word would function as the number-theoretic prime little-omega function LittleOmega(x_{1) }= Sum_{p|X}, where: the summation is taken over all p, such that all of the bound p must be prime, and "|" denotes divisibility of the term on the right (second term) by the term on the left (first term). If X_{2 }= 1, then multiple factors of the same prime are counted (specifically: the (maximal) exponents of the prime factors in the prime factorization of X_{1} are added together); this is the number-theoretic prime big-omega function BigOmega(x_{1) }= Sum_{p^r||X}, where: the notation is as for x_{2 }= -1 or 0 supra as need be, the summation is taken over such p, and "||" denotes the fact that the said corresponding r = v_{p(X} (id est: p^{r} is the maximal power of p which divides X_{1}). No other option for the value of X_{2} is currently defined. X_{2} might have contextual/cultural/conventional defaults, but the contextless default value is X_{2 }= 1. X_{3} specifies the (algebraic) structure in which primehood/factoring is being considered/performed (equipped also with an ordering of the primes); it need not be specified if the context is clear; if such is sensible for the other inputs, the contextless default for X_{3} is the typical ring of integers (with the ordering being the traditional ordering of the integers and the 1st prime being p_{1 }= 2). See also: "pau'au"; https://en.wikipedia.org/wiki/Prime_omega_function .

- pau'au
- ternary mekso operator: p-adic valuation; outputs (positive) infinity if x
_{1 }= 0 and, else, outputs sup(Set(k: k is a nonnegative integer, and ((1 - x_{3)x}divides x_{1))}, where p_{n}is the nth prime (such that p_{1 }= 2). - pi'ei'oi
*(exp!)* - mathematical ternary operator: prime-generating function.