Piles of Plates: S(10^16) mod 1e9+7. O(sqrt(N)) summation: for each k, q = floor((N - k(k-1)/2) / k), contribution = q*(t+1) - k*q*(q+1)/2, where t = N - k(k-1)/2.
# Project Euler 688
# Piles of Plates: S(10^16) mod 1e9+7.
# O(sqrt(N)) summation: for each k, q = floor((N - k(k-1)/2) / k),
# contribution = q*(t+1) - k*q*(q+1)/2, where t = N - k(k-1)/2.
import euler.nt { isqrt }
const MOD: i64 = 1000000007
function main() -> i32 {
let N: i64 = 10000000000000000
let inv2: i64 = (MOD + 1) / 2
# km = floor((sqrt(8N+1) - 1) / 2)
let km: i64 = (isqrt(8 * N + 1) - 1) / 2
let mut t: i64 = N
let mut t_mod: i64 = N % MOD
let mut total: i64 = 0
let mut k: i64 = 1
while k <= km {
let q: i64 = t / k
let q_mod: i64 = q % MOD
# term1 = q * (t+1) mod MOD
let mut x: i64 = t_mod + 1
if x >= MOD { x = x - MOD }
let term1: i128 = (q_mod as i128) * (x as i128) % (MOD as i128)
# term2 = k * q*(q+1)/2 mod MOD
let tri: i128 = (q_mod as i128) * ((q_mod + 1) as i128) % (MOD as i128)
let tri2: i128 = (tri * (inv2 as i128)) % (MOD as i128)
let term2: i128 = ((k as i128) * tri2) % (MOD as i128)
total = total + (term1 as i64)
if total >= MOD { total = total - MOD }
total = total - (term2 as i64)
if total < 0 { total = total + MOD }
t = t - k
t_mod = t_mod - (k % MOD)
if t_mod < 0 { t_mod = t_mod + MOD }
k = k + 1
}
printf("%lld\n", total)
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 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) {
int64_t N = 10000000000000000;
int64_t inv2 = FLOW_CHECKED_DIV(((MOD + 1)), (2));
int64_t km = FLOW_CHECKED_DIV(((isqrt_i64(((8 * N) + 1)) - 1)), (2));
int64_t t = N;
int64_t t_mod = FLOW_CHECKED_MOD((N), (MOD));
int64_t total = 0;
int64_t k = 1;
while (k <= km) {
int64_t q = FLOW_CHECKED_DIV((t), (k));
int64_t q_mod = FLOW_CHECKED_MOD((q), (MOD));
int64_t x = (t_mod + 1);
if (x >= MOD) {
x = (x - MOD);
}
__int128 term1 = FLOW_CHECKED_MOD(((((__int128)(q_mod)) * ((__int128)(x)))), (((__int128)(MOD))));
__int128 tri = FLOW_CHECKED_MOD(((((__int128)(q_mod)) * ((__int128)((q_mod + 1))))), (((__int128)(MOD))));
__int128 tri2 = FLOW_CHECKED_MOD(((tri * ((__int128)(inv2)))), (((__int128)(MOD))));
__int128 term2 = FLOW_CHECKED_MOD(((((__int128)(k)) * tri2)), (((__int128)(MOD))));
total = (total + ((int64_t)(term1)));
if (total >= MOD) {
total = (total - MOD);
}
total = (total - ((int64_t)(term2)));
if (total < 0) {
total = (total + MOD);
}
t = (t - k);
t_mod = (t_mod - FLOW_CHECKED_MOD((k), (MOD)));
if (t_mod < 0) {
t_mod = (t_mod + MOD);
}
k = (k + 1);
}
printf("%lld\n", total);
return 0;
}