# Project Euler 606: Gozinta Chains II
# S(N) = sum of k <= N with g(k)=252, where g(k)=252 iff k=(p*q)^3, p<q prime.
# For N=10^36, M=cbrt(N)=10^12. Compute S mod 10^9.
import euler.nt { isqrt }
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
}
const MOD: i64 = 1000000000
function icbrt(n: i64) -> i64 {
if n < 2 { return n }
let mut hi: i64 = 1
let mut bits: i64 = 0
let mut tmp: i64 = n
while tmp > 0 { bits = bits + 1; tmp = tmp >> 1 }
hi = 1 << ((bits + 2) / 3)
let mut lo: i64 = hi >> 1
while lo + 1 < hi {
let mid: i64 = (lo + hi) >> 1
let m3: i64 = mid * mid * mid
if m3 <= n { lo = mid } else { hi = mid }
}
return lo
}
# Compute (v*(v+1)/2)^2 mod MOD, handling overflow
function sum_cubes_mod(v: i64) -> i64 {
let a: i64
let b: i64
if v % 2 == 0 {
a = v / 2
b = v + 1
} else {
a = v
b = (v + 1) / 2
}
let t: i64 = (a % MOD) * (b % MOD) % MOD
return (t * t - 1) % MOD
}
function main() -> i32 {
let M: i64 = 1000000000000 # 10^12 = cbrt(10^36)
let r: i64 = isqrt(M) # 10^6
# Sieve primes up to r
let bs: ptr<i8> = calloc(r + 1, 1)
for i in 0..(r + 1) { bs[i] = 1 }
bs[0] = 0
bs[1] = 0
let mut p: i64 = 2
while p * p <= r {
if bs[p] == 1 {
let mut j: i64 = p * p
while j <= r {
bs[j] = 0
j = j + p
}
}
p = p + 1
}
# Collect primes
let mut nprimes: i64 = 0
for i in 0..(r + 1) {
if bs[i] == 1 { nprimes = nprimes + 1 }
}
let primes: ptr<i64> = calloc(nprimes, 8)
let mut idx: i64 = 0
for i in 0..(r + 1) {
if bs[i] == 1 {
primes[idx] = i
idx = idx + 1
}
}
# Lucy sieve for prime cube sums
# Key values: V = [M/1, M/2, ..., M/r, r-1, r-2, ..., 1]
let big_len: i64 = r
let small_len: i64 = r - 1
let L: i64 = big_len + small_len # total key values
let V: ptr<i64> = calloc(L, 8)
let S: ptr<i64> = calloc(L, 8)
# Fill V: big block first (decreasing), then small block
for i in 0..big_len {
V[i] = M / (i + 1)
}
let mut vi: i64 = big_len
let mut v: i64 = r - 1
while v >= 1 {
V[vi] = v
vi = vi + 1
v = v - 1
}
# Initialize S[i] = sum_{k=2..V[i]} k^3 mod MOD
for i in 0..L {
S[i] = sum_cubes_mod(V[i])
}
# Index helper: for small x <= r, index is L - x
# For big value M/d (d <= r), index is d - 1
# Sieve
for pi in 0..nprimes {
let p: i64 = primes[pi]
let p2: i64 = p * p
if p2 > M { break }
let p3: i64 = (p * p % MOD) * p % MOD
# sp = sum of prime^3 <= p-1
let sp: i64
if p == 2 {
sp = 0
} else {
sp = S[L - (p - 1)]
}
# Update big block: indices i=0..min(n//p2, r)-1
let mut blen: i64 = M / p2
if blen > r { blen = r }
for i in 0..blen {
let vv: i64 = V[i]
let denom_u: i64 = (i + 1) * p
let Su: i64
if denom_u <= r {
Su = S[denom_u - 1]
} else {
let u: i64 = vv / p
Su = S[L - u]
}
let diff: i64 = (Su - sp) % MOD
S[i] = (S[i] - p3 * diff) % MOD
}
# Update small tail: values from p^2 to r-1
let slen: i64 = r - p2
if slen > 0 {
let start: i64 = r # first tail index in V
let end: i64 = r + slen
for j in start..end {
let vv: i64 = V[j]
let u: i64 = vv / p
let Su: i64 = S[L - u]
let diff: i64 = (Su - sp) % MOD
S[j] = (S[j] - p3 * diff) % MOD
}
}
}
# Compute total
let mut total: i64 = 0
for pi in 0..nprimes {
let p: i64 = primes[pi]
if M / p <= p { break }
# sum of q^3 for primes q with p < q <= M/p
let sum_up_to_qmax: i64 = S[p - 1]
let sum_up_to_p: i64 = S[L - p]
let sum_q: i64 = (sum_up_to_qmax - sum_up_to_p) % MOD
let p3: i64 = (p * p % MOD) * p % MOD
total = (total + p3 * sum_q) % MOD
}
# Ensure positive
total = total % MOD
if total < 0 { total = total + MOD }
printf("%lld\n", total)
free(S)
free(V)
free(primes)
free(bs)
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 icbrt_i64(int64_t n);
int64_t sum_cubes_mod_i64(int64_t v);
int32_t main(void);
static const int64_t MOD = 1000000000;
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 icbrt_i64(int64_t n) {
if (n < 2) {
return n;
}
int64_t hi = 1;
int64_t bits = 0;
int64_t tmp = n;
while (tmp > 0) {
bits = (bits + 1);
tmp = FLOW_CHECKED_SHR((tmp), (1));
}
hi = FLOW_CHECKED_SHL((1), (FLOW_CHECKED_DIV(((bits + 2)), (3))));
int64_t lo = FLOW_CHECKED_SHR((hi), (1));
while ((lo + 1) < hi) {
int64_t mid = FLOW_CHECKED_SHR(((lo + hi)), (1));
int64_t m3 = ((mid * mid) * mid);
if (m3 <= n) {
lo = mid;
} else {
hi = mid;
}
}
return lo;
}
int64_t sum_cubes_mod_i64(int64_t v) {
int64_t a;
int64_t b;
if (FLOW_CHECKED_MOD((v), (2)) == 0) {
a = FLOW_CHECKED_DIV((v), (2));
b = (v + 1);
} else {
a = v;
b = FLOW_CHECKED_DIV(((v + 1)), (2));
}
int64_t t = FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD((a), (MOD)) * FLOW_CHECKED_MOD((b), (MOD)))), (MOD));
return FLOW_CHECKED_MOD((((t * t) - 1)), (MOD));
}
int32_t main(void) {
int64_t M = 1000000000000;
int64_t r = isqrt_i64(M);
int8_t* bs = (int8_t*)(calloc((r + 1), 1));
int32_t __flow_step_1 = 1;
for (int32_t i = 0; (0 <= (r + 1)) ? i < (r + 1) : i > (r + 1); i += (0 <= (r + 1)) ? 1 : -1) {
bs[i] = 1;
}
bs[0] = 0;
bs[1] = 0;
int64_t p = 2;
while ((p * p) <= r) {
if (bs[p] == 1) {
int64_t j = (p * p);
while (j <= r) {
bs[j] = 0;
j = (j + p);
}
}
p = (p + 1);
}
int64_t nprimes = 0;
int32_t __flow_step_2 = 1;
for (int32_t i = 0; (0 <= (r + 1)) ? i < (r + 1) : i > (r + 1); i += (0 <= (r + 1)) ? 1 : -1) {
if (bs[i] == 1) {
nprimes = (nprimes + 1);
}
}
int64_t* primes = (int64_t*)(calloc(nprimes, 8));
int64_t idx = 0;
int32_t __flow_step_3 = 1;
for (int32_t i = 0; (0 <= (r + 1)) ? i < (r + 1) : i > (r + 1); i += (0 <= (r + 1)) ? 1 : -1) {
if (bs[i] == 1) {
primes[idx] = i;
idx = (idx + 1);
}
}
int64_t big_len = r;
int64_t small_len = (r - 1);
int64_t L = (big_len + small_len);
int64_t* V = (int64_t*)(calloc(L, 8));
int64_t* S = (int64_t*)(calloc(L, 8));
int32_t __flow_step_4 = 1;
for (int32_t i = 0; (0 <= big_len) ? i < big_len : i > big_len; i += (0 <= big_len) ? 1 : -1) {
V[i] = FLOW_CHECKED_DIV((M), ((i + 1)));
}
int64_t vi = big_len;
int64_t v = (r - 1);
while (v >= 1) {
V[vi] = v;
vi = (vi + 1);
v = (v - 1);
}
int32_t __flow_step_5 = 1;
for (int32_t i = 0; (0 <= L) ? i < L : i > L; i += (0 <= L) ? 1 : -1) {
S[i] = sum_cubes_mod_i64(V[i]);
}
int32_t __flow_step_6 = 1;
for (int32_t pi = 0; (0 <= nprimes) ? pi < nprimes : pi > nprimes; pi += (0 <= nprimes) ? 1 : -1) {
int64_t p = primes[pi];
int64_t p2 = (p * p);
if (p2 > M) {
break;
}
int64_t p3 = FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD(((p * p)), (MOD)) * p)), (MOD));
int64_t sp;
if (p == 2) {
sp = 0;
} else {
sp = S[(L - (p - 1))];
}
int64_t blen = FLOW_CHECKED_DIV((M), (p2));
if (blen > r) {
blen = r;
}
int32_t __flow_step_7 = 1;
for (int32_t i = 0; (0 <= blen) ? i < blen : i > blen; i += (0 <= blen) ? 1 : -1) {
int64_t vv = V[i];
int64_t denom_u = ((i + 1) * p);
int64_t Su;
if (denom_u <= r) {
Su = S[(denom_u - 1)];
} else {
int64_t u = FLOW_CHECKED_DIV((vv), (p));
Su = S[(L - u)];
}
int64_t diff = FLOW_CHECKED_MOD(((Su - sp)), (MOD));
S[i] = FLOW_CHECKED_MOD(((S[i] - (p3 * diff))), (MOD));
}
int64_t slen = (r - p2);
if (slen > 0) {
int64_t start = r;
int64_t end = (r + slen);
int32_t __flow_step_8 = 1;
for (int32_t j = start; (start <= end) ? j < end : j > end; j += (start <= end) ? 1 : -1) {
int64_t vv = V[j];
int64_t u = FLOW_CHECKED_DIV((vv), (p));
int64_t Su = S[(L - u)];
int64_t diff = FLOW_CHECKED_MOD(((Su - sp)), (MOD));
S[j] = FLOW_CHECKED_MOD(((S[j] - (p3 * diff))), (MOD));
}
}
}
int64_t total = 0;
int32_t __flow_step_9 = 1;
for (int32_t pi = 0; (0 <= nprimes) ? pi < nprimes : pi > nprimes; pi += (0 <= nprimes) ? 1 : -1) {
int64_t p = primes[pi];
if (FLOW_CHECKED_DIV((M), (p)) <= p) {
break;
}
int64_t sum_up_to_qmax = S[(p - 1)];
int64_t sum_up_to_p = S[(L - p)];
int64_t sum_q = FLOW_CHECKED_MOD(((sum_up_to_qmax - sum_up_to_p)), (MOD));
int64_t p3 = FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD(((p * p)), (MOD)) * p)), (MOD));
total = FLOW_CHECKED_MOD(((total + (p3 * sum_q))), (MOD));
}
total = FLOW_CHECKED_MOD((total), (MOD));
if (total < 0) {
total = (total + MOD);
}
printf("%lld\n", total);
free(S);
free(V);
free(primes);
free(bs);
return 0;
}