# Project Euler 378
# Tr(6e7) last 18 digits: decreasing dT triples via Fenwick trees.
import euler.nt { isqrt }
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
function realloc(p: ptr<void>, size: i64) -> ptr<void>
}
function fenwick_sum(bit: ptr<i64>, idx0: i64) -> i64 {
let mut idx: i64 = idx0
let mut s: i64 = 0
while idx > 0 {
s = s + bit[idx]
idx = idx & (idx - 1)
}
return s
}
function fenwick_add(bit: ptr<i64>, size: i64, idx0: i64, delta: i64) -> void {
let mut idx: i64 = idx0
while idx <= size {
bit[idx] = bit[idx] + delta
idx = idx + (idx & (0 - idx))
}
}
function main() -> i32 {
let N: i64 = 60000000
let MOD: i64 = 1000000000000000000
let limit: i64 = N + 1
let spf_odd: ptr<i32> = calloc((limit >> 1) + 1, 4)
let root: i64 = isqrt(limit)
let mut p: i64 = 3
while p <= root {
if spf_odd[p >> 1] == 0 {
let step: i64 = p << 1
let mut m: i64 = p * p
while m <= limit {
let idx: i64 = m >> 1
if spf_odd[idx] == 0 { spf_odd[idx] = p as i32 }
m = m + step
}
}
p = p + 2
}
let tau: ptr<i16> = calloc(limit + 1, 2)
let expv: ptr<i8> = calloc(limit + 1, 1)
tau[1] = 1
let mut x: i64 = 2
while x <= limit {
if (x & 1) == 0 {
let m: i64 = x >> 1
if (m & 1) == 0 {
let e: i64 = (expv[m] as i64) + 1
expv[x] = e as i8
tau[x] = (((tau[m] as i64) / e) * (e + 1)) as i16
} else {
expv[x] = 1
tau[x] = ((tau[m] as i64) * 2) as i16
}
} else {
let sp: i64 = spf_odd[x >> 1] as i64
if sp == 0 {
expv[x] = 1
tau[x] = 2
} else {
let m: i64 = x / sp
if m % sp == 0 {
let e: i64 = (expv[m] as i64) + 1
expv[x] = e as i8
tau[x] = (((tau[m] as i64) / e) * (e + 1)) as i16
} else {
expv[x] = 1
tau[x] = ((tau[m] as i64) * 2) as i16
}
}
}
x = x + 1
}
let mut size: i64 = 1 << 16
let bit1: ptr<i64> = calloc(size + 2, 8)
let bit2: ptr<i64> = calloc(size + 2, 8)
let mut seen: i64 = 0
let mut total_pairs: i64 = 0
let mut ans: i64 = 0
let mut i: i64 = 1
while i <= N {
let mut xv: i64 = 0
if (i & 1) == 0 {
xv = (tau[i >> 1] as i64) * (tau[i + 1] as i64)
} else {
xv = (tau[i] as i64) * (tau[(i + 1) >> 1] as i64)
}
if xv > size {
# grow - realloc not always safe with Flow ptr types; use large enough size
# dT max for this range is well under 2000, so 1<<16 is enough
printf("BIT overflow %lld\n", xv)
return 1
}
let leq: i64 = fenwick_sum(bit1, xv)
let pairs_here: i64 = seen - leq
let pairs_leq: i64 = fenwick_sum(bit2, xv)
let triples_here: i64 = total_pairs - pairs_leq
ans = ans + triples_here
if ans >= MOD { ans = ans % MOD }
if ans < 0 { ans = ans % MOD; if ans < 0 { ans = ans + MOD } }
total_pairs = total_pairs + pairs_here
fenwick_add(bit2, size, xv, pairs_here)
fenwick_add(bit1, size, xv, 1)
seen = seen + 1
i = i + 1
}
printf("%lld\n", ans)
free(bit2)
free(bit1)
free(expv)
free(tau)
free(spf_odd)
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 fenwick_sum_ptr_i64_i64(int64_t* bit, int64_t idx0);
void fenwick_add_ptr_i64_i64_i64_i64(int64_t* bit, int64_t size, int64_t idx0, int64_t delta);
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 fenwick_sum_ptr_i64_i64(int64_t* bit, int64_t idx0) {
int64_t idx = idx0;
int64_t s = 0;
while (idx > 0) {
s = (s + bit[idx]);
idx = (idx & (idx - 1));
}
return s;
}
void fenwick_add_ptr_i64_i64_i64_i64(int64_t* bit, int64_t size, int64_t idx0, int64_t delta) {
int64_t idx = idx0;
while (idx <= size) {
bit[idx] = (bit[idx] + delta);
idx = (idx + (idx & (0 - idx)));
}
}
int32_t main(void) {
int64_t N = 60000000;
int64_t MOD = 1000000000000000000;
int64_t limit = (N + 1);
int32_t* spf_odd = (int32_t*)(calloc((FLOW_CHECKED_SHR((limit), (1)) + 1), 4));
int64_t root = isqrt_i64(limit);
int64_t p = 3;
while (p <= root) {
if (spf_odd[FLOW_CHECKED_SHR((p), (1))] == 0) {
int64_t step = FLOW_CHECKED_SHL((p), (1));
int64_t m = (p * p);
while (m <= limit) {
int64_t idx = FLOW_CHECKED_SHR((m), (1));
if (spf_odd[idx] == 0) {
spf_odd[idx] = ((int32_t)(p));
}
m = (m + step);
}
}
p = (p + 2);
}
int16_t* tau = (int16_t*)(calloc((limit + 1), 2));
int8_t* expv = (int8_t*)(calloc((limit + 1), 1));
tau[1] = 1;
int64_t x = 2;
while (x <= limit) {
if ((x & 1) == 0) {
int64_t m = FLOW_CHECKED_SHR((x), (1));
if ((m & 1) == 0) {
int64_t e = (((int64_t)(expv[m])) + 1);
expv[x] = ((int8_t)(e));
tau[x] = ((int16_t)((FLOW_CHECKED_DIV((((int64_t)(tau[m]))), (e)) * (e + 1))));
} else {
expv[x] = 1;
tau[x] = ((int16_t)((((int64_t)(tau[m])) * 2)));
}
} else {
int64_t sp = ((int64_t)(spf_odd[FLOW_CHECKED_SHR((x), (1))]));
if (sp == 0) {
expv[x] = 1;
tau[x] = 2;
} else {
int64_t m = FLOW_CHECKED_DIV((x), (sp));
if (FLOW_CHECKED_MOD((m), (sp)) == 0) {
int64_t e = (((int64_t)(expv[m])) + 1);
expv[x] = ((int8_t)(e));
tau[x] = ((int16_t)((FLOW_CHECKED_DIV((((int64_t)(tau[m]))), (e)) * (e + 1))));
} else {
expv[x] = 1;
tau[x] = ((int16_t)((((int64_t)(tau[m])) * 2)));
}
}
}
x = (x + 1);
}
int64_t size = FLOW_CHECKED_SHL((1), (16));
int64_t* bit1 = (int64_t*)(calloc((size + 2), 8));
int64_t* bit2 = (int64_t*)(calloc((size + 2), 8));
int64_t seen = 0;
int64_t total_pairs = 0;
int64_t ans = 0;
int64_t i = 1;
while (i <= N) {
int64_t xv = 0;
if ((i & 1) == 0) {
xv = (((int64_t)(tau[FLOW_CHECKED_SHR((i), (1))])) * ((int64_t)(tau[(i + 1)])));
} else {
xv = (((int64_t)(tau[i])) * ((int64_t)(tau[FLOW_CHECKED_SHR(((i + 1)), (1))])));
}
if (xv > size) {
printf("BIT overflow %lld\n", xv);
return 1;
}
int64_t leq = fenwick_sum_ptr_i64_i64(bit1, xv);
int64_t pairs_here = (seen - leq);
int64_t pairs_leq = fenwick_sum_ptr_i64_i64(bit2, xv);
int64_t triples_here = (total_pairs - pairs_leq);
ans = (ans + triples_here);
if (ans >= MOD) {
ans = FLOW_CHECKED_MOD((ans), (MOD));
}
if (ans < 0) {
ans = FLOW_CHECKED_MOD((ans), (MOD));
if (ans < 0) {
ans = (ans + MOD);
}
}
total_pairs = (total_pairs + pairs_here);
fenwick_add_ptr_i64_i64_i64_i64(bit2, size, xv, pairs_here);
fenwick_add_ptr_i64_i64_i64_i64(bit1, size, xv, 1);
seen = (seen + 1);
i = (i + 1);
}
printf("%lld\n", ans);
free(bit2);
free(bit1);
free(expv);
free(tau);
free(spf_odd);
return 0;
}