# Project Euler 379
# g(10^12) = number of pairs x<=y with lcm(x,y)<=N via Lucy / un-Lucy.
import euler.nt { isqrt }
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
}
function lower_bound(arr: ptr<i64>, n: i64, key: i64) -> i64 {
let mut lo: i64 = 0
let mut hi: i64 = n
while lo < hi {
let mid: i64 = (lo + hi) / 2
if arr[mid] < key {
lo = mid + 1
} else {
hi = mid
}
}
return lo
}
function main() -> i32 {
let N: i64 = 1000000000000
let m: i64 = isqrt(N)
# Build key values V = distinct floor(N/i), ascending
let Vtmp: ptr<i64> = calloc(2 * m + 10, 8)
let mut vc: i64 = 0
let mut i: i64 = 1
while i <= N {
let q: i64 = N / i
Vtmp[vc] = q
vc = vc + 1
i = N / q + 1
}
# reverse to ascending
let V: ptr<i64> = calloc(vc, 8)
let mut j: i64 = 0
while j < vc {
V[j] = Vtmp[vc - 1 - j]
j = j + 1
}
free(Vtmp)
let L: i64 = vc
let id2: ptr<i32> = calloc(m + 1, 4)
j = 0
while j <= m {
id2[j] = -1
j = j + 1
}
j = 0
while j < L {
let v: i64 = V[j]
if v > m {
id2[N / v] = j as i32
}
j = j + 1
}
let S: ptr<i64> = calloc(L, 8)
j = 0
while j < L {
S[j] = V[j] - 1
j = j + 1
}
# primes <= m
let sieve: ptr<i8> = calloc(m + 1, 1)
let primes: ptr<i32> = calloc(m / 5 + 10, 4)
let mut pc: i64 = 0
sieve[0] = 1
sieve[1] = 1
i = 2
while i <= m {
if sieve[i] == 0 {
primes[pc] = i as i32
pc = pc + 1
let mut t: i64 = i * i
while t <= m {
sieve[t] = 1
t = t + i
}
}
i = i + 1
}
# Lucy sieve for pi
let mut pi: i64 = 0
while pi < pc {
let p: i64 = primes[pi] as i64
let p2: i64 = p * p
if p2 > N { break }
let sp: i64 = S[p - 2]
let start: i64 = lower_bound(V, L, p2)
j = L - 1
while j >= start {
let v: i64 = V[j]
let u: i64 = v / p
let mut Sj: i64 = 0
if u <= m {
Sj = S[u - 1]
} else {
Sj = S[id2[N / u] as i64]
}
S[j] = S[j] - (Sj - sp)
j = j - 1
}
pi = pi + 1
}
j = 0
while j < L {
S[j] = S[j] * 3
j = j + 1
}
# Un-Lucy
pi = pc - 1
while pi >= 0 {
let p: i64 = primes[pi] as i64
let p2: i64 = p * p
if p2 <= N {
let sp: i64 = S[p - 1]
let start: i64 = lower_bound(V, L, p2)
j = L - 1
while j >= start {
let v: i64 = V[j]
let mut u: i64 = v / p
let mut e: i64 = 1
while u >= p {
let mut Sj: i64 = 0
if u <= m {
Sj = S[u - 1]
} else {
Sj = S[id2[N / u] as i64]
}
S[j] = S[j] + (2 * e + 1) * (Sj - sp) + (2 * (e + 1) + 1)
e = e + 1
u = u / p
}
j = j - 1
}
}
pi = pi - 1
}
let Lsum: i64 = S[L - 1] + 1
let g: i64 = (Lsum + N) / 2
printf("%lld\n", g)
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 lower_bound_ptr_i64_i64_i64(int64_t* arr, int64_t n, int64_t key);
int32_t main(void);
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 lower_bound_ptr_i64_i64_i64(int64_t* arr, int64_t n, int64_t key) {
int64_t lo = 0;
int64_t hi = n;
while (lo < hi) {
int64_t mid = FLOW_CHECKED_DIV(((lo + hi)), (2));
if (arr[mid] < key) {
lo = (mid + 1);
} else {
hi = mid;
}
}
return lo;
}
int32_t main(void) {
int64_t N = 1000000000000;
int64_t m = isqrt_i64(N);
int64_t* Vtmp = (int64_t*)(calloc(((2 * m) + 10), 8));
int64_t vc = 0;
int64_t i = 1;
while (i <= N) {
int64_t q = FLOW_CHECKED_DIV((N), (i));
Vtmp[vc] = q;
vc = (vc + 1);
i = (FLOW_CHECKED_DIV((N), (q)) + 1);
}
int64_t* V = (int64_t*)(calloc(vc, 8));
int64_t j = 0;
while (j < vc) {
V[j] = Vtmp[((vc - 1) - j)];
j = (j + 1);
}
free(Vtmp);
int64_t L = vc;
int32_t* id2 = (int32_t*)(calloc((m + 1), 4));
j = 0;
while (j <= m) {
id2[j] = (-1);
j = (j + 1);
}
j = 0;
while (j < L) {
int64_t v = V[j];
if (v > m) {
id2[FLOW_CHECKED_DIV((N), (v))] = ((int32_t)(j));
}
j = (j + 1);
}
int64_t* S = (int64_t*)(calloc(L, 8));
j = 0;
while (j < L) {
S[j] = (V[j] - 1);
j = (j + 1);
}
int8_t* sieve = (int8_t*)(calloc((m + 1), 1));
int32_t* primes = (int32_t*)(calloc((FLOW_CHECKED_DIV((m), (5)) + 10), 4));
int64_t pc = 0;
sieve[0] = 1;
sieve[1] = 1;
i = 2;
while (i <= m) {
if (sieve[i] == 0) {
primes[pc] = ((int32_t)(i));
pc = (pc + 1);
int64_t t = (i * i);
while (t <= m) {
sieve[t] = 1;
t = (t + i);
}
}
i = (i + 1);
}
int64_t pi = 0;
while (pi < pc) {
int64_t p = ((int64_t)(primes[pi]));
int64_t p2 = (p * p);
if (p2 > N) {
break;
}
int64_t sp = S[(p - 2)];
int64_t start = lower_bound_ptr_i64_i64_i64(V, L, p2);
j = (L - 1);
while (j >= start) {
int64_t v = V[j];
int64_t u = FLOW_CHECKED_DIV((v), (p));
int64_t Sj = 0;
if (u <= m) {
Sj = S[(u - 1)];
} else {
Sj = S[((int64_t)(id2[FLOW_CHECKED_DIV((N), (u))]))];
}
S[j] = (S[j] - (Sj - sp));
j = (j - 1);
}
pi = (pi + 1);
}
j = 0;
while (j < L) {
S[j] = (S[j] * 3);
j = (j + 1);
}
pi = (pc - 1);
while (pi >= 0) {
int64_t p = ((int64_t)(primes[pi]));
int64_t p2 = (p * p);
if (p2 <= N) {
int64_t sp = S[(p - 1)];
int64_t start = lower_bound_ptr_i64_i64_i64(V, L, p2);
j = (L - 1);
while (j >= start) {
int64_t v = V[j];
int64_t u = FLOW_CHECKED_DIV((v), (p));
int64_t e = 1;
while (u >= p) {
int64_t Sj = 0;
if (u <= m) {
Sj = S[(u - 1)];
} else {
Sj = S[((int64_t)(id2[FLOW_CHECKED_DIV((N), (u))]))];
}
S[j] = ((S[j] + (((2 * e) + 1) * (Sj - sp))) + ((2 * (e + 1)) + 1));
e = (e + 1);
u = FLOW_CHECKED_DIV((u), (p));
}
j = (j - 1);
}
}
pi = (pi - 1);
}
int64_t Lsum = (S[(L - 1)] + 1);
int64_t g = FLOW_CHECKED_DIV(((Lsum + N)), (2));
printf("%lld\n", g);
return 0;
}