# Project Euler 650
# Divisors of binomial product: track sigma(B_n) mod 1e9+7 up to n=20000.
import euler.nt { mod_pow }
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
}
const LIMIT: i32 = 20000
const MOD: i64 = 1000000007
function main() -> i32 {
let spf: ptr<i32> = calloc((LIMIT + 1) as i64, 4)
let primes: ptr<i32> = calloc(LIMIT as i64, 4)
let mut pc: i32 = 0
# Linear sieve
let mut x: i32 = 2
while x <= LIMIT {
if spf[x] == 0 {
spf[x] = x
primes[pc] = x
pc = pc + 1
}
let mut j: i32 = 0
while j < pc {
let y: i64 = (primes[j] as i64) * (x as i64)
if y > (LIMIT as i64) { break }
spf[y as i32] = primes[j]
if primes[j] == spf[x] { break }
j = j + 1
}
x = x + 1
}
let pidx: ptr<i32> = calloc((LIMIT + 1) as i64, 4)
let mut i: i32 = 0
while i < pc {
pidx[primes[i]] = i
i = i + 1
}
let prime_power: ptr<i64> = calloc(pc as i64, 8)
let inv_fact: ptr<i64> = calloc(pc as i64, 8)
let inv_prime: ptr<i64> = calloc(pc as i64, 8)
let sigma_den: ptr<i64> = calloc(pc as i64, 8)
let mut k: i32 = 0
while k < pc {
prime_power[k] = (primes[k] as i64) % MOD
inv_fact[k] = 1
inv_prime[k] = mod_pow(primes[k] as i64, MOD - 2, MOD)
sigma_den[k] = mod_pow((primes[k] as i64) - 1, MOD - 2, MOD)
k = k + 1
}
let mut active: i32 = 0
let mut total: i64 = 1
let mut n: i32 = 2
while n <= LIMIT {
while active < pc && primes[active] <= n {
active = active + 1
}
# Divide each active prime_power by its current factorial exponent
let mut a: i32 = 0
while a < active {
prime_power[a] = ((prime_power[a] as i128) * inv_fact[a] % MOD) as i64
a = a + 1
}
# Factor n and update prime_power and inv_fact
let mut xf: i32 = n
while xf > 1 {
let p: i32 = spf[xf]
let mut e: i32 = 0
while xf % p == 0 {
xf = xf / p
e = e + 1
}
let idx: i32 = pidx[p]
let pe: i64 = mod_pow(p as i64, ((n - 1) as i64) * (e as i64), MOD)
prime_power[idx] = ((prime_power[idx] as i128) * pe % MOD) as i64
let ipow: i64 = mod_pow(inv_prime[idx], e as i64, MOD)
inv_fact[idx] = ((inv_fact[idx] as i128) * ipow % MOD) as i64
}
# Compute divisor sum = product of (prime_power[i] - 1) / (prime[i] - 1)
let mut divisor_sum: i64 = 1
let mut b: i32 = 0
while b < active {
let mut t: i64 = (prime_power[b] - 1) % MOD
if t < 0 { t = t + MOD }
divisor_sum = ((divisor_sum as i128) * t % MOD * sigma_den[b] % MOD) as i64
b = b + 1
}
total = (total + divisor_sum) % MOD
n = n + 1
}
printf("%lld\n", total)
free(spf)
free(primes)
free(pidx)
free(prime_power)
free(inv_fact)
free(inv_prime)
free(sigma_den)
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);
int32_t main(void);
static const int32_t LIMIT = 20000;
static const int64_t MOD = 1000000007;
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;
}
int32_t main(void) {
int32_t* spf = (int32_t*)(calloc(((int64_t)((LIMIT + 1))), 4));
int32_t* primes = (int32_t*)(calloc(((int64_t)(LIMIT)), 4));
int32_t pc = 0;
int32_t x = 2;
while (x <= LIMIT) {
if (spf[x] == 0) {
spf[x] = x;
primes[pc] = x;
pc = (pc + 1);
}
int32_t j = 0;
while (j < pc) {
int64_t y = (((int64_t)(primes[j])) * ((int64_t)(x)));
if (y > ((int64_t)(LIMIT))) {
break;
}
spf[((int32_t)(y))] = primes[j];
if (primes[j] == spf[x]) {
break;
}
j = (j + 1);
}
x = (x + 1);
}
int32_t* pidx = (int32_t*)(calloc(((int64_t)((LIMIT + 1))), 4));
int32_t i = 0;
while (i < pc) {
pidx[primes[i]] = i;
i = (i + 1);
}
int64_t* prime_power = (int64_t*)(calloc(((int64_t)(pc)), 8));
int64_t* inv_fact = (int64_t*)(calloc(((int64_t)(pc)), 8));
int64_t* inv_prime = (int64_t*)(calloc(((int64_t)(pc)), 8));
int64_t* sigma_den = (int64_t*)(calloc(((int64_t)(pc)), 8));
int32_t k = 0;
while (k < pc) {
prime_power[k] = FLOW_CHECKED_MOD((((int64_t)(primes[k]))), (MOD));
inv_fact[k] = 1;
inv_prime[k] = mod_pow_i64_i64_i64(((int64_t)(primes[k])), (MOD - 2), MOD);
sigma_den[k] = mod_pow_i64_i64_i64((((int64_t)(primes[k])) - 1), (MOD - 2), MOD);
k = (k + 1);
}
int32_t active = 0;
int64_t total = 1;
int32_t n = 2;
while (n <= LIMIT) {
while ((active < pc && primes[active] <= n)) {
active = (active + 1);
}
int32_t a = 0;
while (a < active) {
prime_power[a] = ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(prime_power[a])) * inv_fact[a])), (MOD))));
a = (a + 1);
}
int32_t xf = n;
while (xf > 1) {
int32_t p = spf[xf];
int32_t e = 0;
while (FLOW_CHECKED_MOD((xf), (p)) == 0) {
xf = FLOW_CHECKED_DIV((xf), (p));
e = (e + 1);
}
int32_t idx = pidx[p];
int64_t pe = mod_pow_i64_i64_i64(((int64_t)(p)), (((int64_t)((n - 1))) * ((int64_t)(e))), MOD);
prime_power[idx] = ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(prime_power[idx])) * pe)), (MOD))));
int64_t ipow = mod_pow_i64_i64_i64(inv_prime[idx], ((int64_t)(e)), MOD);
inv_fact[idx] = ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(inv_fact[idx])) * ipow)), (MOD))));
}
int64_t divisor_sum = 1;
int32_t b = 0;
while (b < active) {
int64_t t = FLOW_CHECKED_MOD(((prime_power[b] - 1)), (MOD));
if (t < 0) {
t = (t + MOD);
}
divisor_sum = ((int64_t)(FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD(((((__int128)(divisor_sum)) * t)), (MOD)) * sigma_den[b])), (MOD))));
b = (b + 1);
}
total = FLOW_CHECKED_MOD(((total + divisor_sum)), (MOD));
n = (n + 1);
}
printf("%lld\n", total);
free(spf);
free(primes);
free(pidx);
free(prime_power);
free(inv_fact);
free(inv_prime);
free(sigma_den);
return 0;
}