From 88d5600bc1b01f84fd40548fb95ddf1660d2f182 Mon Sep 17 00:00:00 2001
From: Horace He <horacehe2007@yahoo.com>
Date: Tue, 14 Apr 2020 03:44:21 -0400
Subject: [PATCH] tried speculative thing

---
 content/number-theory/phiFunction.h | 38 +++++++++++++++--------------
 1 file changed, 20 insertions(+), 18 deletions(-)

diff --git a/content/number-theory/phiFunction.h b/content/number-theory/phiFunction.h
index 7724c3e81..5774b3619 100644
--- a/content/number-theory/phiFunction.h
+++ b/content/number-theory/phiFunction.h
@@ -1,27 +1,29 @@
 /**
- * Author: Håkan Terelius
- * Date: 2009-09-25
+ * Author:
+ * Date:
  * License: CC0
- * Source: http://en.wikipedia.org/wiki/Euler's_totient_function
- * Description: \emph{Euler's $\phi$} function is defined as $\phi(n):=\#$ of positive integers $\leq n$ that are coprime with $n$.
- * $\phi(1)=1$, $p$ prime $\Rightarrow \phi(p^k)=(p-1)p^{k-1}$, $m,n$ coprime $\Rightarrow \phi(mn)=\phi(m)\phi(n)$.
- * If $n=p_1^{k_1}p_2^{k_2} ... p_r^{k_r}$ then $\phi(n) = (p_1-1)p_1^{k_1-1}...(p_r-1)p_r^{k_r-1}$.
- * $\phi(n)=n \cdot \prod_{p|n}(1-1/p)$.
+ * Source:
+ * Description: Multiplicative functions
  *
- * $\sum_{d|n} \phi(d) = n$, $\sum_{1\leq k \leq n, \gcd(k,n)=1} k = n \phi(n)/2, n>1$
- * 
- * \textbf{Euler's thm}: $a,n$ coprime $\Rightarrow a^{\phi(n)} \equiv 1 \pmod{n}$.
+ * Euler's totient: $\phi(1) = 1, \phi(p) = p - 1, \phi(ip) = \phi(i)\phi(p)$
+ *
+ * Mobius function: $\mu(1) = 1, \mu(p) = -1, \mu(ip) = 0$
  *
- * \textbf{Fermat's little thm}: $p$ prime $\Rightarrow a^{p-1} \equiv 1 \pmod{p}$ $\forall a$.
  * Status: Tested
  */
 #pragma once
 
-const int LIM = 5000000;
-int phi[LIM];
-
-void calculatePhi() {
-	rep(i,0,LIM) phi[i] = i&1 ? i : i/2;
-	for(int i = 3; i < LIM; i += 2) if(phi[i] == i)
-		for(int j = i; j < LIM; j += i) phi[j] -= phi[j] / i;
+vector<int> calcMult(int n) {
+	vector<char> sieve(n);
+	vector<int> phi(n), pr; phi[1] = 1; // f(1)
+	rep(i, 2, n) {
+		if (!sieve[i]) pr.push_back(i), phi[i] = i - 1; // f(p)
+		trav(j, pr) {
+			if (i * j >= n) break;
+			sieve[i * j] = true;
+			if (i % j == 0) { phi[i * j] = phi[i] * j; break; } // f(i*p)
+			phi[i * j] = phi[i] * phi[j];
+		}
+	}
+	return phi;
 }