fei'i VUhU3 experimental cmavo

mekso variable-arity (at most ternary) operator: number of prime divisors of number X1, counting with or without multiplicity according to the value X2 (1 xor 0 respectively; see note for equality to -1 and for default value), in structure X3.

X1 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. x2 toggles the type of counting and must be exactly one element of Set(-1, 0, 1). If x2 = -1 and X3 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 vp, where: pi is the ith prime (such that p1 = 2), and vp is the p-adic valuation (see: "pau'au") of the input; in other words, this mode yields the index i of the greatest prime pi which has a nonzero power ri such that pi^r divides X1; if X1 is a unit and x2 = -1, then this word outputs 0; if X1 = 0 and X2 = -1, then this word outputs positive infinity; this mode counts early primes which have power 0 in the prime factorization of X1 but does not count the infinitely many later ones which occur after the last nonzero prime power in that factorization (when X1 is not 0 and is not a unit). If X2 = 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(x1) = Sump|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 X2 = 1, then multiple factors of the same prime are counted (specifically: the (maximal) exponents of the prime factors in the prime factorization of X1 are added together); this is the number-theoretic prime big-omega function BigOmega(x1) = Sump^r||X, where: the notation is as for x2 = -1 or 0 supra as need be, the summation is taken over such p, and "||" denotes the fact that the said corresponding r = vp(X (id est: pr is the maximal power of p which divides X1). No other option for the value of X2 is currently defined. X2 might have contextual/cultural/conventional defaults, but the contextless default value is X2 = 1. X3 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 X3 is the typical ring of integers (with the ordering being the traditional ordering of the integers and the 1st prime being p1 = 2). See also: "pau'au"; https://en.wikipedia.org/wiki/Prime_omega_function .

In notes:

ternary mekso operator: p-adic valuation; outputs (positive) infinity if x1 = 0 and, else, outputs sup(Set(k: k is a nonnegative integer, and ((1 - x3)x divides x1)), where pn is the nth prime (such that p1 = 2).
pi'ei'oi (exp!)
mathematical ternary operator: prime-generating function.