# Project Euler 319
# t(10^10) mod 10^9 via Möbius / Mertens + geometric G.
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
}
function icbrt(n: i64) -> i64 {
if n <= 0 { return 0 }
let mut x: i64 = 1
while x * x * x <= n { x = x * 2 }
let mut lo: i64 = 0
let mut hi: i64 = x
while lo + 1 < hi {
let mid: i64 = (lo + hi) / 2
if mid * mid * mid <= n { lo = mid } else { hi = mid }
}
return lo
}
function modpow(base: i64, exp: i64, mod: i64) -> i64 {
let mut r: i64 = 1
let mut b: i64 = ((base % mod) + mod) % mod
let mut e: i64 = exp
while e > 0 {
if e % 2 == 1 { r = (r * b) % mod }
b = (b * b) % mod
e = e / 2
}
return r
}
function slot(n: i64, keys: ptr<i64>, used: ptr<i8>, cap: i64) -> i64 {
let mut h: i64 = n % cap
if h < 0 { h = h + cap }
while used[h] == 1 && keys[h] != n {
h = h + 1
if h == cap { h = 0 }
}
return h
}
function M(n: i64, pref: ptr<i32>, lim: i64, keys: ptr<i64>, vals: ptr<i64>, used: ptr<i8>, cap: i64) -> i64
function M(n: i64, pref: ptr<i32>, lim: i64, keys: ptr<i64>, vals: ptr<i64>, used: ptr<i8>, cap: i64) -> i64 {
if n <= 0 { return 0 }
if n <= lim { return pref[n] as i64 }
let s: i64 = slot(n, keys, used, cap)
if used[s] == 1 { return vals[s] }
let mut res: i64 = 1
let mut i: i64 = 2
while i <= n {
let q: i64 = n / i
let j: i64 = n / q
res = res - (j - i + 1) * M(q, pref, lim, keys, vals, used, cap)
i = j + 1
}
used[s] = 1
keys[s] = n
vals[s] = res
return res
}
function G_mod(m: i64, mod: i64) -> i64 {
if m <= 0 { return 0 }
let mod2: i64 = 2 * mod
let num: i64 = (modpow(3, m + 1, mod2) - 3 + mod2) % mod2
let sum3: i64 = (num / 2) % mod
let sum2: i64 = (modpow(2, m + 1, mod) - 2 + mod) % mod
return (sum3 - sum2 - (m % mod) + 2 * mod) % mod
}
function main() -> i32 {
let N: i64 = 10000000000
let mod: i64 = 1000000000
let c: i64 = icbrt(N)
let lim: i64 = c * c
let is_comp: ptr<i8> = calloc(lim + 1, 1)
let mu: ptr<i8> = calloc(lim + 1, 1)
let primes: ptr<i32> = calloc(lim / 5 + 10, 4)
mu[1] = 1
let mut pc: i64 = 0
let mut i: i64 = 2
while i <= lim {
if is_comp[i] == 0 {
primes[pc] = i as i32
pc = pc + 1
mu[i] = -1
}
let mut j: i64 = 0
while j < pc {
let p: i64 = primes[j] as i64
let ip: i64 = i * p
if ip > lim { break }
is_comp[ip] = 1
if i % p == 0 {
mu[ip] = 0
break
}
mu[ip] = (0 - (mu[i] as i64)) as i8
j = j + 1
}
i = i + 1
}
let pref: ptr<i32> = calloc(lim + 1, 4)
let mut s: i64 = 0
i = 1
while i <= lim {
s = s + (mu[i] as i64)
pref[i] = s as i32
i = i + 1
}
let cap: i64 = 2097152
let keys: ptr<i64> = calloc(cap, 8)
let vals: ptr<i64> = calloc(cap, 8)
let used: ptr<i8> = calloc(cap, 1)
let mut ans: i64 = 0
i = 1
while i <= N {
let q: i64 = N / i
let j: i64 = N / q
let mu_sum: i64 = M(j, pref, lim, keys, vals, used, cap) - M(i - 1, pref, lim, keys, vals, used, cap)
ans = (ans + (mu_sum % mod + mod) % mod * G_mod(q, mod)) % mod
i = j + 1
}
printf("%lld\n", (ans + 1) % mod)
free(is_comp); free(mu); free(primes); free(pref); free(keys); free(vals); free(used)
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 icbrt_i64(int64_t n);
int64_t modpow_i64_i64_i64(int64_t base, int64_t exp, int64_t mod);
int64_t slot_i64_ptr_i64_ptr_i8_i64(int64_t n, int64_t* keys, int8_t* used, int64_t cap);
int64_t M_i64_ptr_i32_i64_ptr_i64_ptr_i64_ptr_i8_i64(int64_t n, int32_t* pref, int64_t lim, int64_t* keys, int64_t* vals, int8_t* used, int64_t cap);
int64_t M_i64_ptr_i32_i64_ptr_i64_ptr_i64_ptr_i8_i64(int64_t n, int32_t* pref, int64_t lim, int64_t* keys, int64_t* vals, int8_t* used, int64_t cap);
int64_t G_mod_i64_i64(int64_t m, int64_t mod);
int32_t main(void);
int64_t icbrt_i64(int64_t n) {
if (n <= 0) {
return 0;
}
int64_t x = 1;
while (((x * x) * x) <= n) {
x = (x * 2);
}
int64_t lo = 0;
int64_t hi = x;
while ((lo + 1) < hi) {
int64_t mid = FLOW_CHECKED_DIV(((lo + hi)), (2));
if (((mid * mid) * mid) <= n) {
lo = mid;
} else {
hi = mid;
}
}
return lo;
}
int64_t modpow_i64_i64_i64(int64_t base, int64_t exp, int64_t mod) {
int64_t r = 1;
int64_t b = FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD((base), (mod)) + mod)), (mod));
int64_t e = exp;
while (e > 0) {
if (FLOW_CHECKED_MOD((e), (2)) == 1) {
r = FLOW_CHECKED_MOD(((r * b)), (mod));
}
b = FLOW_CHECKED_MOD(((b * b)), (mod));
e = FLOW_CHECKED_DIV((e), (2));
}
return r;
}
int64_t slot_i64_ptr_i64_ptr_i8_i64(int64_t n, int64_t* keys, int8_t* used, int64_t cap) {
int64_t h = FLOW_CHECKED_MOD((n), (cap));
if (h < 0) {
h = (h + cap);
}
while ((used[h] == 1 && keys[h] != n)) {
h = (h + 1);
if (h == cap) {
h = 0;
}
}
return h;
}
int64_t M_i64_ptr_i32_i64_ptr_i64_ptr_i64_ptr_i8_i64(int64_t n, int32_t* pref, int64_t lim, int64_t* keys, int64_t* vals, int8_t* used, int64_t cap) {
if (n <= 0) {
return 0;
}
if (n <= lim) {
return ((int64_t)(pref[n]));
}
int64_t s = slot_i64_ptr_i64_ptr_i8_i64(n, keys, used, cap);
if (used[s] == 1) {
return vals[s];
}
int64_t res = 1;
int64_t i = 2;
while (i <= n) {
int64_t q = FLOW_CHECKED_DIV((n), (i));
int64_t j = FLOW_CHECKED_DIV((n), (q));
res = (res - (((j - i) + 1) * M_i64_ptr_i32_i64_ptr_i64_ptr_i64_ptr_i8_i64(q, pref, lim, keys, vals, used, cap)));
i = (j + 1);
}
used[s] = 1;
keys[s] = n;
vals[s] = res;
return res;
}
int64_t G_mod_i64_i64(int64_t m, int64_t mod) {
if (m <= 0) {
return 0;
}
int64_t mod2 = (2 * mod);
int64_t num = FLOW_CHECKED_MOD((((modpow_i64_i64_i64(3, (m + 1), mod2) - 3) + mod2)), (mod2));
int64_t sum3 = FLOW_CHECKED_MOD((FLOW_CHECKED_DIV((num), (2))), (mod));
int64_t sum2 = FLOW_CHECKED_MOD((((modpow_i64_i64_i64(2, (m + 1), mod) - 2) + mod)), (mod));
return FLOW_CHECKED_MOD(((((sum3 - sum2) - FLOW_CHECKED_MOD((m), (mod))) + (2 * mod))), (mod));
}
int32_t main(void) {
int64_t N = 10000000000;
int64_t mod = 1000000000;
int64_t c = icbrt_i64(N);
int64_t lim = (c * c);
int8_t* is_comp = (int8_t*)(calloc((lim + 1), 1));
int8_t* mu = (int8_t*)(calloc((lim + 1), 1));
int32_t* primes = (int32_t*)(calloc((FLOW_CHECKED_DIV((lim), (5)) + 10), 4));
mu[1] = 1;
int64_t pc = 0;
int64_t i = 2;
while (i <= lim) {
if (is_comp[i] == 0) {
primes[pc] = ((int32_t)(i));
pc = (pc + 1);
mu[i] = (-1);
}
int64_t j = 0;
while (j < pc) {
int64_t p = ((int64_t)(primes[j]));
int64_t ip = (i * p);
if (ip > lim) {
break;
}
is_comp[ip] = 1;
if (FLOW_CHECKED_MOD((i), (p)) == 0) {
mu[ip] = 0;
break;
}
mu[ip] = ((int8_t)((0 - ((int64_t)(mu[i])))));
j = (j + 1);
}
i = (i + 1);
}
int32_t* pref = (int32_t*)(calloc((lim + 1), 4));
int64_t s = 0;
i = 1;
while (i <= lim) {
s = (s + ((int64_t)(mu[i])));
pref[i] = ((int32_t)(s));
i = (i + 1);
}
int64_t cap = 2097152;
int64_t* keys = (int64_t*)(calloc(cap, 8));
int64_t* vals = (int64_t*)(calloc(cap, 8));
int8_t* used = (int8_t*)(calloc(cap, 1));
int64_t ans = 0;
i = 1;
while (i <= N) {
int64_t q = FLOW_CHECKED_DIV((N), (i));
int64_t j = FLOW_CHECKED_DIV((N), (q));
int64_t mu_sum = (M_i64_ptr_i32_i64_ptr_i64_ptr_i64_ptr_i8_i64(j, pref, lim, keys, vals, used, cap) - M_i64_ptr_i32_i64_ptr_i64_ptr_i64_ptr_i8_i64((i - 1), pref, lim, keys, vals, used, cap));
ans = FLOW_CHECKED_MOD(((ans + (FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD((mu_sum), (mod)) + mod)), (mod)) * G_mod_i64_i64(q, mod)))), (mod));
i = (j + 1);
}
printf("%lld\n", FLOW_CHECKED_MOD(((ans + 1)), (mod)));
free(is_comp);
free(mu);
free(primes);
free(pref);
free(keys);
free(vals);
free(used);
return 0;
}