# Project Euler 492
# Exploding Sequence — B(10^9, 10^7, 10^15).
import euler.nt { isqrt }
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
}
let mut P_LUCAS: i64 = 11
function modpow(base0: i64, exp0: i64, mod: i64) -> i64 {
let mut r: i64 = 1
let mut b: i64 = base0 % mod
let mut e: i64 = exp0
if b < 0 { b = b + mod }
while e > 0 {
if e % 2 == 1 {
r = (r * b) % mod
}
b = (b * b) % mod
e = e / 2
}
return r
}
function bit_length(n0: i64) -> i64 {
let mut n: i64 = n0
let mut k: i64 = 0
while n > 0 {
n = n / 2
k = k + 1
}
return k
}
# Return U_n mod mod for Lucas(P=11, Q=1); also writes U_{n+1} into up1_out[0].
function lucas_U_pair(n0: i64, mod: i64, up1_out: ptr<i64>) -> i64 {
if n0 == 0 {
up1_out[0] = 1
return 0
}
let mut u: i64 = 0
let mut up1: i64 = 1
let mut mask: i64 = 1
let bl: i64 = bit_length(n0)
let mut s: i64 = 1
while s < bl {
mask = mask * 2
s = s + 1
}
# mask = 2^(bit_length-1)
while mask > 0 {
let mut v: i64 = (2 * up1 - P_LUCAS * u) % mod
if v < 0 { v = v + mod }
let u2k: i64 = (u * v) % mod
let mut u2k1: i64 = (up1 * v - 1) % mod
if u2k1 < 0 { u2k1 = u2k1 + mod }
if (n0 & mask) != 0 {
let mut u2k2: i64 = (P_LUCAS * u2k1 - u2k) % mod
if u2k2 < 0 { u2k2 = u2k2 + mod }
u = u2k1
up1 = u2k2
} else {
u = u2k
up1 = u2k1
}
mask = mask / 2
}
up1_out[0] = up1
return u
}
function lucas_V(n: i64, mod: i64) -> i64 {
let up1_out: ptr<i64> = calloc(1, 8)
let u: i64 = lucas_U_pair(n, mod, up1_out)
let up1: i64 = up1_out[0]
free(up1_out)
let mut r: i64 = (2 * up1 - P_LUCAS * u) % mod
if r < 0 { r = r + mod }
return r
}
function residue13(p: i64) -> bool {
let r: i64 = p % 13
if r == 1 { return true }
if r == 3 { return true }
if r == 4 { return true }
if r == 9 { return true }
if r == 10 { return true }
if r == 12 { return true }
return false
}
function a_mod_prime(n: i64, p: i64) -> i64 {
if p == 2 {
return 1 % p
}
if p == 3 {
let mut a: i64 = 1 % p
let mut t: i64 = 1
while t < n {
a = (6 * a * a + 10 * a + 3) % p
t = t + 1
}
return a
}
let mut leg: i64 = 0 - 1
if residue13(p) {
leg = 1
}
let mm: i64 = p - leg
let e: i64 = modpow(2, n - 1, mm)
let u_n: i64 = lucas_V(e, p)
let inv6: i64 = modpow(6, p - 2, p)
let mut r: i64 = ((u_n - 5) % p) * inv6 % p
if r < 0 { r = r + p }
return r
}
function main() -> i32 {
let X: i64 = 1000000000
let Y: i64 = 10000000
let N: i64 = 1000000000000000
let high: i64 = X + Y
let base_limit: i64 = isqrt(high) + 1
let sieve: ptr<i8> = calloc(base_limit + 1, 1)
if sieve == null { return 1 }
let mut i: i64 = 0
while i <= base_limit {
sieve[i] = 1
i = i + 1
}
if base_limit >= 0 { sieve[0] = 0 }
if base_limit >= 1 { sieve[1] = 0 }
i = 2
while i * i <= base_limit {
if sieve[i] == 1 {
let mut j: i64 = i * i
while j <= base_limit {
sieve[j] = 0
j = j + i
}
}
i = i + 1
}
let mut nbase: i64 = 0
i = 2
while i <= base_limit {
if sieve[i] == 1 { nbase = nbase + 1 }
i = i + 1
}
let base_primes: ptr<i64> = calloc(nbase, 8)
if base_primes == null { return 1 }
let mut bi: i64 = 0
i = 2
while i <= base_limit {
if sieve[i] == 1 {
base_primes[bi] = i
bi = bi + 1
}
i = i + 1
}
# Segmented sieve over [X, X+Y], odd-only.
let mut start: i64 = X
if (start & 1) == 0 { start = start + 1 }
let seg_size: i64 = ((high - start) / 2) + 1
let seg: ptr<i8> = calloc(seg_size, 1)
if seg == null { return 1 }
i = 0
while i < seg_size {
seg[i] = 1
i = i + 1
}
bi = 0
while bi < nbase {
let q: i64 = base_primes[bi]
if q == 2 {
bi = bi + 1
continue
}
let qq: i64 = q * q
if qq > high { break }
let mut first: i64 = 0
if qq >= start {
first = qq
} else {
first = ((start + q - 1) / q) * q
}
if (first & 1) == 0 { first = first + q }
let mut idx: i64 = (first - start) / 2
while idx < seg_size {
seg[idx] = 0
idx = idx + q
}
bi = bi + 1
}
let mut total: i64 = 0
if X <= 2 {
if 2 <= high {
total = total + a_mod_prime(N, 2)
}
}
i = 0
while i < seg_size {
if seg[i] == 1 {
let p: i64 = start + 2 * i
if p >= X {
if p <= high {
total = total + a_mod_prime(N, p)
}
}
}
i = i + 1
}
printf("%lld\n", total)
free(sieve)
free(base_primes)
free(seg)
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 modpow_i64_i64_i64(int64_t base0, int64_t exp0, int64_t mod);
int64_t bit_length_i64(int64_t n0);
int64_t lucas_U_pair_i64_i64_ptr_i64(int64_t n0, int64_t mod, int64_t* up1_out);
int64_t lucas_V_i64_i64(int64_t n, int64_t mod);
bool residue13_i64(int64_t p);
int64_t a_mod_prime_i64_i64(int64_t n, int64_t p);
int32_t main(void);
/* Module statics */
static int64_t P_LUCAS = 11;
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 modpow_i64_i64_i64(int64_t base0, int64_t exp0, int64_t mod) {
int64_t r = 1;
int64_t b = FLOW_CHECKED_MOD((base0), (mod));
int64_t e = exp0;
if (b < 0) {
b = (b + mod);
}
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 bit_length_i64(int64_t n0) {
int64_t n = n0;
int64_t k = 0;
while (n > 0) {
n = FLOW_CHECKED_DIV((n), (2));
k = (k + 1);
}
return k;
}
int64_t lucas_U_pair_i64_i64_ptr_i64(int64_t n0, int64_t mod, int64_t* up1_out) {
if (n0 == 0) {
up1_out[0] = 1;
return 0;
}
int64_t u = 0;
int64_t up1 = 1;
int64_t mask = 1;
int64_t bl = bit_length_i64(n0);
int64_t s = 1;
while (s < bl) {
mask = (mask * 2);
s = (s + 1);
}
while (mask > 0) {
int64_t v = FLOW_CHECKED_MOD((((2 * up1) - (P_LUCAS * u))), (mod));
if (v < 0) {
v = (v + mod);
}
int64_t u2k = FLOW_CHECKED_MOD(((u * v)), (mod));
int64_t u2k1 = FLOW_CHECKED_MOD((((up1 * v) - 1)), (mod));
if (u2k1 < 0) {
u2k1 = (u2k1 + mod);
}
if ((n0 & mask) != 0) {
int64_t u2k2 = FLOW_CHECKED_MOD((((P_LUCAS * u2k1) - u2k)), (mod));
if (u2k2 < 0) {
u2k2 = (u2k2 + mod);
}
u = u2k1;
up1 = u2k2;
} else {
u = u2k;
up1 = u2k1;
}
mask = FLOW_CHECKED_DIV((mask), (2));
}
up1_out[0] = up1;
return u;
}
int64_t lucas_V_i64_i64(int64_t n, int64_t mod) {
int64_t* up1_out = (int64_t*)(calloc(1, 8));
int64_t u = lucas_U_pair_i64_i64_ptr_i64(n, mod, up1_out);
int64_t up1 = up1_out[0];
free(up1_out);
int64_t r = FLOW_CHECKED_MOD((((2 * up1) - (P_LUCAS * u))), (mod));
if (r < 0) {
r = (r + mod);
}
return r;
}
bool residue13_i64(int64_t p) {
int64_t r = FLOW_CHECKED_MOD((p), (13));
if (r == 1) {
return 1;
}
if (r == 3) {
return 1;
}
if (r == 4) {
return 1;
}
if (r == 9) {
return 1;
}
if (r == 10) {
return 1;
}
if (r == 12) {
return 1;
}
return 0;
}
int64_t a_mod_prime_i64_i64(int64_t n, int64_t p) {
if (p == 2) {
return FLOW_CHECKED_MOD((1), (p));
}
if (p == 3) {
int64_t a = FLOW_CHECKED_MOD((1), (p));
int64_t t = 1;
while (t < n) {
a = FLOW_CHECKED_MOD((((((6 * a) * a) + (10 * a)) + 3)), (p));
t = (t + 1);
}
return a;
}
int64_t leg = (0 - 1);
if (residue13_i64(p)) {
leg = 1;
}
int64_t mm = (p - leg);
int64_t e = modpow_i64_i64_i64(2, (n - 1), mm);
int64_t u_n = lucas_V_i64_i64(e, p);
int64_t inv6 = modpow_i64_i64_i64(6, (p - 2), p);
int64_t r = FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD(((u_n - 5)), (p)) * inv6)), (p));
if (r < 0) {
r = (r + p);
}
return r;
}
int32_t main(void) {
int64_t X = 1000000000;
int64_t Y = 10000000;
int64_t N = 1000000000000000;
int64_t high = (X + Y);
int64_t base_limit = (isqrt_i64(high) + 1);
int8_t* sieve = (int8_t*)(calloc((base_limit + 1), 1));
if (sieve == NULL) {
return 1;
}
int64_t i = 0;
while (i <= base_limit) {
sieve[i] = 1;
i = (i + 1);
}
if (base_limit >= 0) {
sieve[0] = 0;
}
if (base_limit >= 1) {
sieve[1] = 0;
}
i = 2;
while ((i * i) <= base_limit) {
if (sieve[i] == 1) {
int64_t j = (i * i);
while (j <= base_limit) {
sieve[j] = 0;
j = (j + i);
}
}
i = (i + 1);
}
int64_t nbase = 0;
i = 2;
while (i <= base_limit) {
if (sieve[i] == 1) {
nbase = (nbase + 1);
}
i = (i + 1);
}
int64_t* base_primes = (int64_t*)(calloc(nbase, 8));
if (base_primes == NULL) {
return 1;
}
int64_t bi = 0;
i = 2;
while (i <= base_limit) {
if (sieve[i] == 1) {
base_primes[bi] = i;
bi = (bi + 1);
}
i = (i + 1);
}
int64_t start = X;
if ((start & 1) == 0) {
start = (start + 1);
}
int64_t seg_size = (FLOW_CHECKED_DIV(((high - start)), (2)) + 1);
int8_t* seg = (int8_t*)(calloc(seg_size, 1));
if (seg == NULL) {
return 1;
}
i = 0;
while (i < seg_size) {
seg[i] = 1;
i = (i + 1);
}
bi = 0;
while (bi < nbase) {
int64_t q = base_primes[bi];
if (q == 2) {
bi = (bi + 1);
continue;
}
int64_t qq = (q * q);
if (qq > high) {
break;
}
int64_t first = 0;
if (qq >= start) {
first = qq;
} else {
first = (FLOW_CHECKED_DIV((((start + q) - 1)), (q)) * q);
}
if ((first & 1) == 0) {
first = (first + q);
}
int64_t idx = FLOW_CHECKED_DIV(((first - start)), (2));
while (idx < seg_size) {
seg[idx] = 0;
idx = (idx + q);
}
bi = (bi + 1);
}
int64_t total = 0;
if (X <= 2) {
if (2 <= high) {
total = (total + a_mod_prime_i64_i64(N, 2));
}
}
i = 0;
while (i < seg_size) {
if (seg[i] == 1) {
int64_t p = (start + (2 * i));
if (p >= X) {
if (p <= high) {
total = (total + a_mod_prime_i64_i64(N, p));
}
}
}
i = (i + 1);
}
printf("%lld\n", total);
free(sieve);
free(base_primes);
free(seg);
return 0;
}