Problem 69: Totient maximum

eu069.c

@@ -0,0 +1,63 @@
+#include "euler.h"
+
+static int primes[1000000];
+static int nprimes;
+static int cachedphi[1000000];
+
+static int phi(int n) {
+  int p;
+  for (int i = 0; i < nprimes && ((p = primes[i]) < n); i++) {
+    if (n % p == 0) {
+      // If p and n/p are coprime, we can multiply them,
+      // otherwise we have to cast out factors of p.
+      if (n % (p*p) != 0) {
+        return (cachedphi[n] = cachedphi[n / p] * cachedphi[p]);
+      } else {
+        int k = 2;
+        int divisor = p*p;
+        do {
+          k++;
+          divisor *= p;
+        } while (n % divisor == 0);
+        divisor /= p;
+
+        // phi(p^k) = p^k-p^{k-1} and phi(n) = phi(n/p^k) * phi(p^k)
+        int phi_p_k = divisor - divisor / p;
+
+        return (cachedphi[n] = cachedphi[n / divisor] * phi_p_k);
+      }
+    }
+  }
+
+  // phi(p) = p-1 when p is prime
+  return (cachedphi[n] = n-1);
+}
+
+static double n_over_phi_n(int n) {
+  double d = n;
+  d /= phi(n);
+  return d;
+}
+
+void eu069(char *ans) {
+  const int N = 1000000;
+  /* const int N = 20; */
+
+  double max = 0;
+  int n_of_max = 0;
+
+  nprimes = genprimes(primes, N);
+
+  cachedphi[1] = 1;
+
+  for (int n = 2; n < N; n++) {
+    double v = n_over_phi_n(n);
+    /* printf("%d: %d %g\n", n, phi(n), v); */
+    if (v > max) {
+      max = v;
+      n_of_max = n;
+    }
+  }
+
+  sprintf(ans, "%d", n_of_max);
+}

euler.c

@@ -253,6 +253,7 @@ struct puzzle puzzles[] = {
   { "066", &eu066, "661" },
   { "067", &eu067, "7273" },
   { "068", &eu068, "6531031914842725" },
+  { "069", &eu069, NULL },
 };
 
 #define NPUZZLES (sizeof puzzles / sizeof(puzzles[0]))