Return-path: Envelope-to: jbovlaste-admin@lojban.org Delivery-date: Wed, 26 Oct 2022 22:29:19 -0700 Received: from [192.168.123.254] (port=36944 helo=jiten.lojban.org) by d7893716a6e6 with smtp (Exim 4.94.2) (envelope-from ) id 1onvS8-007EtI-P1 for jbovlaste-admin@lojban.org; Wed, 26 Oct 2022 22:29:19 -0700 Received: by jiten.lojban.org (sSMTP sendmail emulation); Thu, 27 Oct 2022 05:29:16 +0000 From: "Apache" To: curtis289@att.net Reply-To: webmaster@lojban.org Subject: [jvsw] Definition Edited At Word fei'i -- By krtisfranks Date: Thu, 27 Oct 2022 05:29:16 +0000 MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Message-Id: X-Spam-Score: -2.9 (--) X-Spam_score: -2.9 X-Spam_score_int: -28 X-Spam_bar: -- In jbovlaste, the user krtisfranks has edited a definition of "fei'i" in the language "English". Differences: 11a12,14 \n> Word: prime omega function, In Sense: little omega > Word: prime omega function, In Sense: big omega > Word: prime-factor-counting function, In Sense: Old Data: Definition: 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 $-1$), in structure $X_3$. Notes: $X_1$ 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 $x_2 = -1$, then the output is $k =$ sup$($Set$(i: i$ is a positive integer, and $v_{p_i}(X_1) > 0)))$, where: $p_i$ is the $i$th prime (if $X_3$ is the ring of integers, then the ordering here is the traditional ordering of the integers and $p_1 = 2$) if such makes sense, 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 $p_i^{r_i}$ 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 $\omega(x_1) =$ Sum$_{p|x_1} (1)$, 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 (the maximal exponents of the prime factors in the prime factorization of $X_1$ are added together); this the number-theoretic prime big-omega function $\omega(x_1) =$ Sum$_{p^r||x_1} (1)$, where: the notation is as for $x_2 = 0$ and the summation is taken over such $p$, except "||" denotes the fact that the said corresponding $r = v_p(X_1)$. 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. See also: "{pau'au}"; https://en.wikipedia.org/wiki/Prime_omega_function . Jargon: Gloss Keywords: Word: prime factors count, In Sense: Place Keywords: New Data: Definition: 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 $-1$), in structure $X_3$. Notes: $X_1$ 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 $x_2 = -1$, then the output is $k =$ sup$($Set$(i: i$ is a positive integer, and $v_{p_i}(X_1) > 0)))$, where: $p_i$ is the $i$th prime (if $X_3$ is the ring of integers, then the ordering here is the traditional ordering of the integers and $p_1 = 2$) if such makes sense, 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 $p_i^{r_i}$ 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 $\omega(x_1) =$ Sum$_{p|x_1} (1)$, 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 (the maximal exponents of the prime factors in the prime factorization of $X_1$ are added together); this the number-theoretic prime big-omega function $\omega(x_1) =$ Sum$_{p^r||x_1} (1)$, where: the notation is as for $x_2 = 0$ and the summation is taken over such $p$, except "||" denotes the fact that the said corresponding $r = v_p(X_1)$. 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. See also: "{pau'au}"; https://en.wikipedia.org/wiki/Prime_omega_function . Jargon: Gloss Keywords: Word: prime factors count, In Sense: Word: prime omega function, In Sense: little omega Word: prime omega function, In Sense: big omega Word: prime-factor-counting function, In Sense: Place Keywords: You can go to to see it.