# Project Euler 484
# Arithmetic Derivative
# S(N) = sum_{1 < k <= N} gcd(k, k') via powerful-number DFS.
import euler.nt { isqrt }
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
}
let mut PRIMES: ptr<i64> = null
let mut PLEN: i64 = 0
function dfs(i0: i64, L0: i64) -> i64 {
let mut res: i64 = 0
let mut i: i64 = i0
while i < PLEN {
let p: i64 = PRIMES[i]
let q: i64 = p * p
let mut L: i64 = L0 / q
if L == 0 { break }
let mut e: i64 = 1
let mut g: i64 = 1
while L != 0 {
let gp: i64 = g
e = e + 1
if e != 1 {
if e == p {
g = g * q
e = 0
} else {
g = g * p
}
let c: i64 = g - gp
res = res + c * L
if L > q {
res = res + c * dfs(i + 1, L)
}
}
L = L / p
}
i = i + 1
}
return res
}
function main() -> i32 {
let N: i64 = 5000000000000000
let lim: i64 = isqrt(N)
let half: i64 = lim / 2 + 1
let sieve: ptr<i8> = calloc(half, 1)
if sieve == null { return 1 }
let mut i: i64 = 0
while i < half {
sieve[i] = 1
i = i + 1
}
sieve[0] = 0
let r: i64 = isqrt(lim)
let mut p: i64 = 3
while p <= r {
if sieve[p / 2] != 0 {
let mut cur: i64 = (p * p) / 2
while cur < half {
sieve[cur] = 0
cur = cur + p
}
}
p = p + 2
}
PRIMES = calloc(4200000, 8)
if PRIMES == null { return 1 }
PRIMES[0] = 2
PLEN = 1
i = 1
while i < half {
if sieve[i] != 0 {
PRIMES[PLEN] = 2 * i + 1
PLEN = PLEN + 1
}
i = i + 1
}
free(sieve)
let ans: i64 = N - 1 + dfs(0, N)
printf("%lld\n", ans)
free(PRIMES)
PRIMES = null
PLEN = 0
return 0
}
Generated C
#include <stdint.h>
#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
/* Flow runtime helpers */
typedef struct flow_temp_node { struct flow_temp_node* next; } flow_temp_node;
static flow_temp_node* flow_temp_head = NULL;
static int flow_temp_atexit_set = 0;
__attribute__((unused)) static void flow_temp_free_all(void) {
while (flow_temp_head) {
flow_temp_node* n = flow_temp_head;
flow_temp_head = n->next;
free(n);
}
}
__attribute__((unused)) static void* flow_temp_alloc(size_t nbytes) {
flow_temp_node* node = (flow_temp_node*)malloc(sizeof(flow_temp_node) + nbytes);
if (!node) return NULL;
node->next = flow_temp_head;
flow_temp_head = node;
if (!flow_temp_atexit_set) {
flow_temp_atexit_set = 1;
atexit(flow_temp_free_all);
}
return (void*)(node + 1);
}
#ifndef FLOW_DIAG
#define FLOW_DIAG(msg) fprintf(stderr, "%s", (msg))
#endif
#ifndef FLOW_LOG
#define FLOW_LOG(fmt, ...) printf(fmt, __VA_ARGS__)
#endif
#ifndef FLOW_LOG_EMPTY
#define FLOW_LOG_EMPTY(fmt) printf(fmt)
#endif
static char* flow_strcat(const char* a, const char* b) {
size_t la = strlen(a ? a : ""), lb = strlen(b ? b : "");
char* r = (char*)flow_temp_alloc(la + lb + 1);
if (!r) return NULL;
if (la) memcpy(r, a, la);
if (lb) memcpy(r + la, b, lb);
r[la + lb] = '\0';
return r;
}
#define __flow_in_arr(arr, val) __extension__ ({ \
int _found = 0; \
size_t _n = sizeof(arr)/sizeof((arr)[0]); \
for (size_t _i = 0; _i < _n; _i++) { \
if ((arr)[_i] == (val)) { _found = 1; break; } \
} _found; })
/* Unified fault handler (MISRA #279) — override with -DFLOW_FAULT_HANDLER=fn */
#ifndef FLOW_FAULT_HANDLER
__attribute__((unused)) static inline void flow_fault_handler(const char* msg) {
fprintf(stderr, "flow: %s\n", msg ? msg : "fault");
abort();
#if defined(__GNUC__) || defined(__clang__)
__builtin_unreachable();
#endif
}
#else
#define flow_fault_handler FLOW_FAULT_HANDLER
#endif
#define flow_div_by_zero_handler() flow_fault_handler("division by zero")
#define flow_shift_ub_handler() flow_fault_handler("invalid shift (amount out of range or left-shift of negative)")
#ifndef FLOW_CHECKED_DIV
#define FLOW_CHECKED_DIV(L, R) (((R) != 0) ? ((L) / (R)) : (flow_div_by_zero_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_MOD
#define FLOW_CHECKED_MOD(L, R) (((R) != 0) ? ((L) % (R)) : (flow_div_by_zero_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_SHL
#define FLOW_CHECKED_SHL(L, R) ((((R) >= 0) && ((unsigned long long)(R) < (sizeof(L) * 8ull)) && ((L) >= 0)) ? ((L) << (R)) : (flow_shift_ub_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_SHR
#define FLOW_CHECKED_SHR(L, R) ((((R) >= 0) && ((unsigned long long)(R) < (sizeof(L) * 8ull))) ? ((L) >> (R)) : (flow_shift_ub_handler(), (L) * 0))
#endif
#include <math.h>
void* _ui_state = NULL;
static inline float i32_to_f32(int32_t v) { return (float)v; }
/* Host stub for @gpu kernels (device codegen replaces this). */
static inline int32_t gpu_thread_id(void) { return 0; }
int64_t gcd_i64_i64(int64_t a0, int64_t b0);
int64_t lcm_i64_i64(int64_t a, int64_t b);
int64_t isqrt_i64(int64_t n);
int64_t mulmod_i64_i64_i64(int64_t a0, int64_t b0, int64_t mod);
int64_t mod_pow_i64_i64_i64(int64_t base, int64_t exp, int64_t mod);
bool is_prime_i64(int64_t n);
int64_t dfs_i64_i64(int64_t i0, int64_t L0);
int32_t main(void);
/* Module statics */
static int64_t* PRIMES = NULL;
static int64_t PLEN = 0;
int64_t gcd_i64_i64(int64_t a0, int64_t b0) {
int64_t a = a0;
int64_t b = b0;
while (b != 0) {
int64_t t = FLOW_CHECKED_MOD((a), (b));
a = b;
b = t;
}
return a;
}
int64_t lcm_i64_i64(int64_t a, int64_t b) {
if ((a == 0 || b == 0)) {
return 0;
}
return (FLOW_CHECKED_DIV((a), (gcd_i64_i64(a, b))) * b);
}
int64_t isqrt_i64(int64_t n) {
if (n < 2) {
return n;
}
int64_t x = n;
int64_t y = FLOW_CHECKED_DIV(((x + 1)), (2));
while (y < x) {
x = y;
y = FLOW_CHECKED_DIV(((x + FLOW_CHECKED_DIV((n), (x)))), (2));
}
return x;
}
int64_t mulmod_i64_i64_i64(int64_t a0, int64_t b0, int64_t mod) {
int64_t a = FLOW_CHECKED_MOD((a0), (mod));
int64_t b = FLOW_CHECKED_MOD((b0), (mod));
int64_t result = 0;
while (b > 0) {
if (FLOW_CHECKED_MOD((b), (2)) == 1) {
result = FLOW_CHECKED_MOD(((result + a)), (mod));
}
a = FLOW_CHECKED_MOD(((a * 2)), (mod));
b = FLOW_CHECKED_DIV((b), (2));
}
return result;
}
int64_t mod_pow_i64_i64_i64(int64_t base, int64_t exp, int64_t mod) {
if (mod == 1) {
return 0;
}
int64_t result = 1;
int64_t b = FLOW_CHECKED_MOD((base), (mod));
int64_t e = exp;
while (e > 0) {
if (FLOW_CHECKED_MOD((e), (2)) == 1) {
result = mulmod_i64_i64_i64(result, b, mod);
}
b = mulmod_i64_i64_i64(b, b, mod);
e = FLOW_CHECKED_DIV((e), (2));
}
return result;
}
bool is_prime_i64(int64_t n) {
if (n < 2) {
return 0;
}
if (n < 4) {
return 1;
}
if ((FLOW_CHECKED_MOD((n), (2)) == 0 || FLOW_CHECKED_MOD((n), (3)) == 0)) {
return 0;
}
int64_t i = 5;
while ((i * i) <= n) {
if ((FLOW_CHECKED_MOD((n), (i)) == 0 || FLOW_CHECKED_MOD((n), ((i + 2))) == 0)) {
return 0;
}
i = (i + 6);
}
return 1;
}
int64_t dfs_i64_i64(int64_t i0, int64_t L0) {
int64_t res = 0;
int64_t i = i0;
while (i < PLEN) {
int64_t p = PRIMES[i];
int64_t q = (p * p);
int64_t L = FLOW_CHECKED_DIV((L0), (q));
if (L == 0) {
break;
}
int64_t e = 1;
int64_t g = 1;
while (L != 0) {
int64_t gp = g;
e = (e + 1);
if (e != 1) {
if (e == p) {
g = (g * q);
e = 0;
} else {
g = (g * p);
}
int64_t c = (g - gp);
res = (res + (c * L));
if (L > q) {
res = (res + (c * dfs_i64_i64((i + 1), L)));
}
}
L = FLOW_CHECKED_DIV((L), (p));
}
i = (i + 1);
}
return res;
}
int32_t main(void) {
int64_t N = 5000000000000000;
int64_t lim = isqrt_i64(N);
int64_t half = (FLOW_CHECKED_DIV((lim), (2)) + 1);
int8_t* sieve = (int8_t*)(calloc(half, 1));
if (sieve == NULL) {
return 1;
}
int64_t i = 0;
while (i < half) {
sieve[i] = 1;
i = (i + 1);
}
sieve[0] = 0;
int64_t r = isqrt_i64(lim);
int64_t p = 3;
while (p <= r) {
if (sieve[FLOW_CHECKED_DIV((p), (2))] != 0) {
int64_t cur = FLOW_CHECKED_DIV(((p * p)), (2));
while (cur < half) {
sieve[cur] = 0;
cur = (cur + p);
}
}
p = (p + 2);
}
PRIMES = calloc(4200000, 8);
if (PRIMES == NULL) {
return 1;
}
PRIMES[0] = 2;
PLEN = 1;
i = 1;
while (i < half) {
if (sieve[i] != 0) {
PRIMES[PLEN] = ((2 * i) + 1);
PLEN = (PLEN + 1);
}
i = (i + 1);
}
free(sieve);
int64_t ans = ((N - 1) + dfs_i64_i64(0, N));
printf("%lld\n", ans);
free(PRIMES);
PRIMES = NULL;
PLEN = 0;
return 0;
}