S(N) = sum_{n<=N} 2^{Omega(n)}, computed for N = 10^14. Identity: 2^{Omega(n)} = sum_{d|n, d powerful} g(d), where a powerful number d = prod p_i^{e_i} (e_i >= 2) carries weight g(d) = prod 2^{e_i - 2}. Then S(N) = sum_{d powerful, d<=N} g(d) * D(N/d), where D(x) = sum_{k<=x} tau(k) = #{(a,b): a*b <= x} = 2*sum_{i<=s} floor(x/i) - s^2, s = isqrt(x). Small x use a precomputed prefix of tau; large x are cached.
# Project Euler 708: Twos Are All You Need
# S(N) = sum_{n<=N} 2^{Omega(n)}, computed for N = 10^14.
#
# Identity: 2^{Omega(n)} = sum_{d|n, d powerful} g(d), where a powerful number
# d = prod p_i^{e_i} (e_i >= 2) carries weight g(d) = prod 2^{e_i - 2}.
# Then S(N) = sum_{d powerful, d<=N} g(d) * D(N/d), where
# D(x) = sum_{k<=x} tau(k) = #{(a,b): a*b <= x} = 2*sum_{i<=s} floor(x/i) - s^2,
# s = isqrt(x). Small x use a precomputed prefix of tau; large x are cached.
import euler.nt { isqrt }
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
function malloc(n: i64) -> ptr<void>
function memset(dst: ptr<void>, val: i32, n: i64) -> ptr<void>
function sqrt(x: f64) -> f64
}
const N_VAL: i64 = 100000000000000
const SMALL_LIMIT: i64 = 1000000
const CACHE_BITS: i64 = 17
const CACHE_SIZE: i64 = 131072
const CACHE_MASK: i64 = 131071
let mut g_primes: ptr<i32> = null
let mut g_prime_sq: ptr<i64> = null
let mut g_nprimes: i64 = 0
let mut g_prefix: ptr<i64> = null
let mut g_N: i64 = 0
let mut g_total: i64 = 0
let mut cache_key: ptr<i64> = null
let mut cache_val: ptr<i64> = null
let mut cache_used: ptr<i8> = null
function isqrt_i64(n: i64) -> i64 {
if n <= 0 { return 0 }
let mut x: i64 = sqrt(n as f64) as i64
while x > 0 && x * x > n {
x = x - 1
}
while (x + 1) * (x + 1) <= n {
x = x + 1
}
return x
}
# Odd-only sieve; returns malloc'd prime list (2, 3, 5, ...).
function primes_upto(limit: i64) -> ptr<i32> {
if limit < 2 { return null }
let size: i64 = limit / 2 + 1
let sieve: ptr<i8> = malloc(size) as ptr<i8>
memset(sieve, 1, size)
sieve[0] = 0
let r: i64 = isqrt_i64(limit)
let mut p: i64 = 3
while p <= r {
if sieve[p / 2] != 0 {
let start: i64 = p * p
let mut j: i64 = start
while j <= limit {
sieve[j / 2] = 0
j = j + 2 * p
}
}
p = p + 2
}
let mut cnt: i64 = 1
let mut i: i64 = 1
while i < size {
if sieve[i] != 0 { cnt = cnt + 1 }
i = i + 1
}
let primes: ptr<i32> = malloc(cnt * 4) as ptr<i32>
primes[0] = 2
let mut k: i64 = 1
i = 1
while i < size {
if sieve[i] != 0 {
primes[k] = (2 * i + 1) as i32
k = k + 1
}
i = i + 1
}
free(sieve)
g_nprimes = cnt
return primes
}
# Prefix sums of tau(k) (number of divisors) for k = 1..limit.
function build_divisor_prefix(limit: i64) -> ptr<i64> {
let spf: ptr<i32> = calloc(limit + 1, 4) as ptr<i32>
let mut i: i64 = 2
while i <= limit {
if spf[i] == 0 {
spf[i] = i as i32
if i * i <= limit {
let mut j: i64 = i * i
while j <= limit {
if spf[j] == 0 { spf[j] = i as i32 }
j = j + i
}
}
}
i = i + 1
}
let d: ptr<i32> = calloc(limit + 1, 4) as ptr<i32>
let exp: ptr<i32> = calloc(limit + 1, 4) as ptr<i32>
d[1] = 1
i = 2
while i <= limit {
let p: i32 = spf[i]
let m: i64 = i / (p as i64)
if m % (p as i64) == 0 {
exp[i] = exp[m] + 1
d[i] = d[m] / (exp[m] + 1) * (exp[i] + 1)
} else {
exp[i] = 1
d[i] = d[m] * 2
}
i = i + 1
}
let prefix: ptr<i64> = calloc(limit + 1, 8) as ptr<i64>
let mut total: i64 = 0
i = 1
while i <= limit {
total = total + (d[i] as i64)
prefix[i] = total
i = i + 1
}
free(spf)
free(d)
free(exp)
return prefix
}
function D_func(x: i64) -> i64 {
if x <= SMALL_LIMIT { return g_prefix[x] }
# Open-addressing hash table lookup
let h: i64 = (x * 11400714819323198485) >> (64 - CACHE_BITS)
let mut idx: i64 = h & CACHE_MASK
while cache_used[idx] != 0 {
if cache_key[idx] == x { return cache_val[idx] }
idx = (idx + 1) & CACHE_MASK
}
let s: i64 = isqrt_i64(x)
let mut total: i64 = 0
let mut i: i64 = 1
while i <= s {
let q: i64 = x / i
let j: i64 = x / q
let jj: i64 = if j > s { s } else { j }
total = total + q * (jj - i + 1)
i = jj + 1
}
let res: i64 = 2 * total - s * s
cache_used[idx] = 1
cache_key[idx] = x
cache_val[idx] = res
return res
}
function dfs(start_idx: i64, current_n: i64, current_g: i64) -> void {
g_total = g_total + current_g * D_func(g_N / current_n)
let limit: i64 = g_N / current_n
let mut i: i64 = start_idx
while i < g_nprimes {
let p2: i64 = g_prime_sq[i]
if p2 > limit {
i = g_nprimes
} else {
let p: i64 = g_primes[i] as i64
let mut pow_p: i64 = p2
let mut g: i64 = current_g
while pow_p <= limit {
dfs(i + 1, current_n * pow_p, g)
if pow_p > limit / p {
pow_p = limit + 1
} else {
pow_p = pow_p * p
g = g << 1
}
}
i = i + 1
}
}
}
function main() -> i32 {
let prime_limit: i64 = isqrt_i64(N_VAL)
g_primes = primes_upto(prime_limit)
g_prime_sq = malloc(g_nprimes * 8) as ptr<i64>
let mut i: i64 = 0
while i < g_nprimes {
let p: i64 = g_primes[i] as i64
g_prime_sq[i] = p * p
i = i + 1
}
g_prefix = build_divisor_prefix(SMALL_LIMIT)
cache_key = calloc(CACHE_SIZE, 8) as ptr<i64>
cache_val = calloc(CACHE_SIZE, 8) as ptr<i64>
cache_used = calloc(CACHE_SIZE, 1) as ptr<i8>
g_N = N_VAL
g_total = 0
dfs(0, 1, 1)
printf("%lld\n", g_total)
free(g_primes)
free(g_prime_sq)
free(g_prefix)
free(cache_key)
free(cache_val)
free(cache_used)
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 isqrt_i64_i64(int64_t n);
int32_t* primes_upto_i64(int64_t limit);
int64_t* build_divisor_prefix_i64(int64_t limit);
int64_t D_func_i64(int64_t x);
void dfs_i64_i64_i64(int64_t start_idx, int64_t current_n, int64_t current_g);
int32_t main(void);
static const int64_t N_VAL = 100000000000000;
static const int64_t SMALL_LIMIT = 1000000;
static const int64_t CACHE_BITS = 17;
static const int64_t CACHE_SIZE = 131072;
static const int64_t CACHE_MASK = 131071;
/* Module statics */
static int32_t* g_primes = NULL;
static int64_t* g_prime_sq = NULL;
static int64_t g_nprimes = 0;
static int64_t* g_prefix = NULL;
static int64_t g_N = 0;
static int64_t g_total = 0;
static int64_t* cache_key = NULL;
static int64_t* cache_val = NULL;
static int8_t* cache_used = NULL;
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 isqrt_i64_i64(int64_t n) {
if (n <= 0) {
return 0;
}
int64_t x = ((int64_t)(sqrt(((double)(n)))));
while ((x > 0 && (x * x) > n)) {
x = (x - 1);
}
while (((x + 1) * (x + 1)) <= n) {
x = (x + 1);
}
return x;
}
int32_t* primes_upto_i64(int64_t limit) {
if (limit < 2) {
return NULL;
}
int64_t size = (FLOW_CHECKED_DIV((limit), (2)) + 1);
int8_t* sieve = (int8_t*)(((int8_t*)(malloc(size))));
memset(sieve, 1, size);
sieve[0] = 0;
int64_t r = isqrt_i64_i64(limit);
int64_t p = 3;
while (p <= r) {
if (sieve[FLOW_CHECKED_DIV((p), (2))] != 0) {
int64_t start = (p * p);
int64_t j = start;
while (j <= limit) {
sieve[FLOW_CHECKED_DIV((j), (2))] = 0;
j = (j + (2 * p));
}
}
p = (p + 2);
}
int64_t cnt = 1;
int64_t i = 1;
while (i < size) {
if (sieve[i] != 0) {
cnt = (cnt + 1);
}
i = (i + 1);
}
int32_t* primes = (int32_t*)(((int32_t*)(malloc((cnt * 4)))));
primes[0] = 2;
int64_t k = 1;
i = 1;
while (i < size) {
if (sieve[i] != 0) {
primes[k] = ((int32_t)(((2 * i) + 1)));
k = (k + 1);
}
i = (i + 1);
}
free(sieve);
g_nprimes = cnt;
return primes;
}
int64_t* build_divisor_prefix_i64(int64_t limit) {
int32_t* spf = (int32_t*)(((int32_t*)(calloc((limit + 1), 4))));
int64_t i = 2;
while (i <= limit) {
if (spf[i] == 0) {
spf[i] = ((int32_t)(i));
if ((i * i) <= limit) {
int64_t j = (i * i);
while (j <= limit) {
if (spf[j] == 0) {
spf[j] = ((int32_t)(i));
}
j = (j + i);
}
}
}
i = (i + 1);
}
int32_t* d = (int32_t*)(((int32_t*)(calloc((limit + 1), 4))));
int32_t* exp = (int32_t*)(((int32_t*)(calloc((limit + 1), 4))));
d[1] = 1;
i = 2;
while (i <= limit) {
int32_t p = spf[i];
int64_t m = FLOW_CHECKED_DIV((i), (((int64_t)(p))));
if (FLOW_CHECKED_MOD((m), (((int64_t)(p)))) == 0) {
exp[i] = (exp[m] + 1);
d[i] = (FLOW_CHECKED_DIV((d[m]), ((exp[m] + 1))) * (exp[i] + 1));
} else {
exp[i] = 1;
d[i] = (d[m] * 2);
}
i = (i + 1);
}
int64_t* prefix = (int64_t*)(((int64_t*)(calloc((limit + 1), 8))));
int64_t total = 0;
i = 1;
while (i <= limit) {
total = (total + ((int64_t)(d[i])));
prefix[i] = total;
i = (i + 1);
}
free(spf);
free(d);
free(exp);
return prefix;
}
int64_t D_func_i64(int64_t x) {
if (x <= SMALL_LIMIT) {
return g_prefix[x];
}
int64_t h = FLOW_CHECKED_SHR(((x * ((__int128)0x9E3779B97F4A7C15ULL))), ((64 - CACHE_BITS)));
int64_t idx = (h & CACHE_MASK);
while (cache_used[idx] != 0) {
if (cache_key[idx] == x) {
return cache_val[idx];
}
idx = ((idx + 1) & CACHE_MASK);
}
int64_t s = isqrt_i64_i64(x);
int64_t total = 0;
int64_t i = 1;
while (i <= s) {
int64_t q = FLOW_CHECKED_DIV((x), (i));
int64_t j = FLOW_CHECKED_DIV((x), (q));
int64_t jj = ((j > s) ? (s) : (j));
total = (total + (q * ((jj - i) + 1)));
i = (jj + 1);
}
int64_t res = ((2 * total) - (s * s));
cache_used[idx] = 1;
cache_key[idx] = x;
cache_val[idx] = res;
return res;
}
void dfs_i64_i64_i64(int64_t start_idx, int64_t current_n, int64_t current_g) {
g_total = (g_total + (current_g * D_func_i64(FLOW_CHECKED_DIV((g_N), (current_n)))));
int64_t limit = FLOW_CHECKED_DIV((g_N), (current_n));
int64_t i = start_idx;
while (i < g_nprimes) {
int64_t p2 = g_prime_sq[i];
if (p2 > limit) {
i = g_nprimes;
} else {
int64_t p = ((int64_t)(g_primes[i]));
int64_t pow_p = p2;
int64_t g = current_g;
while (pow_p <= limit) {
dfs_i64_i64_i64((i + 1), (current_n * pow_p), g);
if (pow_p > FLOW_CHECKED_DIV((limit), (p))) {
pow_p = (limit + 1);
} else {
pow_p = (pow_p * p);
g = FLOW_CHECKED_SHL((g), (1));
}
}
i = (i + 1);
}
}
}
int32_t main(void) {
int64_t prime_limit = isqrt_i64_i64(N_VAL);
g_primes = primes_upto_i64(prime_limit);
g_prime_sq = ((int64_t*)(malloc((g_nprimes * 8))));
int64_t i = 0;
while (i < g_nprimes) {
int64_t p = ((int64_t)(g_primes[i]));
g_prime_sq[i] = (p * p);
i = (i + 1);
}
g_prefix = build_divisor_prefix_i64(SMALL_LIMIT);
cache_key = ((int64_t*)(calloc(CACHE_SIZE, 8)));
cache_val = ((int64_t*)(calloc(CACHE_SIZE, 8)));
cache_used = ((int8_t*)(calloc(CACHE_SIZE, 1)));
g_N = N_VAL;
g_total = 0;
dfs_i64_i64_i64(0, 1, 1);
printf("%lld\n", g_total);
free(g_primes);
free(g_prime_sq);
free(g_prefix);
free(cache_key);
free(cache_val);
free(cache_used);
return 0;
}