# Project Euler 447
# F(10^14) mod 10^9+7 where F(N)=sum_{n=2..N} R(n), R(n)=sigma*(n)-n
# (unitary divisor sum). Uses Möbius inversion:
# sum_{n<=N} sigma*(n) = sum_e μ(e)*e * sum_{k<=N/e^2} k*floor((N/e^2)/k).
import euler.nt { isqrt }
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
}
function sum_k_floor(M: i64, MOD: i64, INV2: i64) -> i64 {
let mut s: i64 = 0
let mut k: i64 = 1
while k <= M {
let q: i64 = M / k
let mut maxk: i64 = M / q
if maxk > M { maxk = M }
let hi: i64 = maxk
let lo: i64 = k
let mut sumk: i64 = ((hi % MOD) * ((hi + 1) % MOD) % MOD) * INV2 % MOD
let lom1: i64 = lo - 1
let sub: i64 = ((lom1 % MOD) * ((lom1 + 1) % MOD) % MOD) * INV2 % MOD
sumk = (sumk - sub) % MOD
if sumk < 0 { sumk = sumk + MOD }
s = (s + (q % MOD) * sumk) % MOD
k = maxk + 1
}
return s
}
function main() -> i32 {
let MOD: i64 = 1000000007
let INV2: i64 = (MOD + 1) / 2
let N: i64 = 100000000000000
let L: i64 = isqrt(N)
let mu: ptr<i8> = calloc(L + 1, 1)
let is_comp: ptr<i8> = calloc(L + 1, 1)
let primes: ptr<i32> = calloc(L / 5 + 10, 4)
let pref: ptr<i64> = calloc(L + 1, 8)
if mu == null || is_comp == null || primes == null || pref == null { return 1 }
mu[1] = 1
let mut pc: i64 = 0
let mut i: i64 = 2
while i <= L {
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 > L { 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
}
i = 1
while i <= L {
let term: i64 = (mu[i] as i64) * (i % MOD) % MOD
pref[i] = (pref[i - 1] + term) % MOD
if pref[i] < 0 { pref[i] = pref[i] + MOD }
i = i + 1
}
# cache for sum_k_floor(M): open-address hash
let HCAP: i64 = 1 << 20
let keys: ptr<i64> = calloc(HCAP, 8)
let vals: ptr<i64> = calloc(HCAP, 8)
let used: ptr<i8> = calloc(HCAP, 1)
if keys == null || vals == null || used == null { return 1 }
let mut total: i64 = 0
let mut e: i64 = 1
while e <= L {
let M: i64 = N / (e * e)
if M == 0 { break }
# largest e2 with N/(e2*e2) == M
let mut e2: i64 = isqrt(N / M)
if e2 > L { e2 = L }
if e2 < e { e2 = e }
while e2 > e && N / (e2 * e2) < M {
e2 = e2 - 1
}
while e2 + 1 <= L && N / ((e2 + 1) * (e2 + 1)) == M {
e2 = e2 + 1
}
let mut sum_emu: i64 = (pref[e2] - pref[e - 1]) % MOD
if sum_emu < 0 { sum_emu = sum_emu + MOD }
# cached sum_k_floor(M)
let mut h: i64 = M % HCAP
let mut piece: i64 = -1
while used[h] != 0 {
if keys[h] == M {
piece = vals[h]
break
}
h = h + 1
if h >= HCAP { h = 0 }
}
if piece < 0 {
piece = sum_k_floor(M, MOD, INV2)
used[h] = 1
keys[h] = M
vals[h] = piece
}
total = (total + sum_emu * piece) % MOD
e = e2 + 1
}
let tri: i64 = ((N % MOD) * ((N + 1) % MOD) % MOD) * INV2 % MOD
let ans: i64 = (total - tri) % MOD
if ans < 0 { ans = ans + MOD }
printf("%lld\n", ans)
free(used)
free(vals)
free(keys)
free(pref)
free(primes)
free(is_comp)
free(mu)
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 sum_k_floor_i64_i64_i64(int64_t M, int64_t MOD, int64_t INV2);
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 sum_k_floor_i64_i64_i64(int64_t M, int64_t MOD, int64_t INV2) {
int64_t s = 0;
int64_t k = 1;
while (k <= M) {
int64_t q = FLOW_CHECKED_DIV((M), (k));
int64_t maxk = FLOW_CHECKED_DIV((M), (q));
if (maxk > M) {
maxk = M;
}
int64_t hi = maxk;
int64_t lo = k;
int64_t sumk = FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD((hi), (MOD)) * FLOW_CHECKED_MOD(((hi + 1)), (MOD)))), (MOD)) * INV2)), (MOD));
int64_t lom1 = (lo - 1);
int64_t sub = FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD((lom1), (MOD)) * FLOW_CHECKED_MOD(((lom1 + 1)), (MOD)))), (MOD)) * INV2)), (MOD));
sumk = FLOW_CHECKED_MOD(((sumk - sub)), (MOD));
if (sumk < 0) {
sumk = (sumk + MOD);
}
s = FLOW_CHECKED_MOD(((s + (FLOW_CHECKED_MOD((q), (MOD)) * sumk))), (MOD));
k = (maxk + 1);
}
return s;
}
int32_t main(void) {
int64_t MOD = 1000000007;
int64_t INV2 = FLOW_CHECKED_DIV(((MOD + 1)), (2));
int64_t N = 100000000000000;
int64_t L = isqrt_i64(N);
int8_t* mu = (int8_t*)(calloc((L + 1), 1));
int8_t* is_comp = (int8_t*)(calloc((L + 1), 1));
int32_t* primes = (int32_t*)(calloc((FLOW_CHECKED_DIV((L), (5)) + 10), 4));
int64_t* pref = (int64_t*)(calloc((L + 1), 8));
if ((((mu == NULL || is_comp == NULL) || primes == NULL) || pref == NULL)) {
return 1;
}
mu[1] = 1;
int64_t pc = 0;
int64_t i = 2;
while (i <= L) {
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 > L) {
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);
}
i = 1;
while (i <= L) {
int64_t term = FLOW_CHECKED_MOD(((((int64_t)(mu[i])) * FLOW_CHECKED_MOD((i), (MOD)))), (MOD));
pref[i] = FLOW_CHECKED_MOD(((pref[(i - 1)] + term)), (MOD));
if (pref[i] < 0) {
pref[i] = (pref[i] + MOD);
}
i = (i + 1);
}
int64_t HCAP = FLOW_CHECKED_SHL((1), (20));
int64_t* keys = (int64_t*)(calloc(HCAP, 8));
int64_t* vals = (int64_t*)(calloc(HCAP, 8));
int8_t* used = (int8_t*)(calloc(HCAP, 1));
if (((keys == NULL || vals == NULL) || used == NULL)) {
return 1;
}
int64_t total = 0;
int64_t e = 1;
while (e <= L) {
int64_t M = FLOW_CHECKED_DIV((N), ((e * e)));
if (M == 0) {
break;
}
int64_t e2 = isqrt_i64(FLOW_CHECKED_DIV((N), (M)));
if (e2 > L) {
e2 = L;
}
if (e2 < e) {
e2 = e;
}
while ((e2 > e && FLOW_CHECKED_DIV((N), ((e2 * e2))) < M)) {
e2 = (e2 - 1);
}
while (((e2 + 1) <= L && FLOW_CHECKED_DIV((N), (((e2 + 1) * (e2 + 1)))) == M)) {
e2 = (e2 + 1);
}
int64_t sum_emu = FLOW_CHECKED_MOD(((pref[e2] - pref[(e - 1)])), (MOD));
if (sum_emu < 0) {
sum_emu = (sum_emu + MOD);
}
int64_t h = FLOW_CHECKED_MOD((M), (HCAP));
int64_t piece = (-1);
while (used[h] != 0) {
if (keys[h] == M) {
piece = vals[h];
break;
}
h = (h + 1);
if (h >= HCAP) {
h = 0;
}
}
if (piece < 0) {
piece = sum_k_floor_i64_i64_i64(M, MOD, INV2);
used[h] = 1;
keys[h] = M;
vals[h] = piece;
}
total = FLOW_CHECKED_MOD(((total + (sum_emu * piece))), (MOD));
e = (e2 + 1);
}
int64_t tri = FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD((N), (MOD)) * FLOW_CHECKED_MOD(((N + 1)), (MOD)))), (MOD)) * INV2)), (MOD));
int64_t ans = FLOW_CHECKED_MOD(((total - tri)), (MOD));
if (ans < 0) {
ans = (ans + MOD);
}
printf("%lld\n", ans);
free(used);
free(vals);
free(keys);
free(pref);
free(primes);
free(is_comp);
free(mu);
return 0;
}