Lucky Numbers
Problem Statement :
A number is called lucky if the sum of its digits, as well as the sum of the squares of its digits is a prime number. How many numbers between a and b inclusive, are lucky?
For example, a=20 and b=25. Each number is tested below:
digit digit squares
value sum squares sum
20 2 4,0 4
21 3 4,1 5
22 4 4,4 8
23 5 4,9 13
24 6 4,16 20
25 7 4,25 29
We see that two numbers, 21, 23 and 25 are lucky.
Note: These lucky numbers are not to be confused with Lucky Numbers
Function Description
Complete the luckyNumbers function in the editor below. It should return an integer that represents the number of lucky numbers in the given range.
luckyNumbers has the following parameter(s):
a: an integer, the lower range bound
b: an integer, the higher range bound
Input Format
The first line contains the number of test cases T.
Each of the next T lines contains two space-separated integers, a and b.
Constraints
1 <= T <= 10^4
1 <= a <= b <= 10^18
Output Format
Output T lines, one for each test case in the order given.
Solution :
Solution in C :
In C++ :
#include <iostream>
#include <cstring>
using namespace std;
typedef long long s64;
const int D = 18;
const int MD = 9;
const s64 MAX = 1000000000000000000ull;
s64 dp[D][D*MD+1][D*MD*MD+1];
int hi[D];
bool isPrime[D*MD*MD+1];
s64 go(int idx, int sd, int ssd, bool limited) {
if (idx == D) return isPrime[sd] && isPrime[ssd] ? 1 : 0;
s64& ret = dp[idx][sd][ssd];
if (!limited && ret >= 0) return ret;
int upper = (limited ? hi[idx] : 9);
s64 r = 0;
for (int i = 0; i <= upper; i++)
r += go(idx+1, sd+i, ssd+i*i, (limited && i == upper));
if (!limited) ret = r;
return r;
}
s64 go(s64 n) {
if (n == MAX) return go(MAX-1);
if (n <= 1) return 0;
for (int i = D-1; i >= 0; i--) hi[i] = (int)(n % 10), n /= 10;
return go(0, 0, 0, true);
}
int main() {
for (int i = 2; i <= D*MD*MD; i++) isPrime[i] = true;
for (int i = 2; i <= D*MD*MD; i++)
if (isPrime[i]) for (int j = i+i; j <= D*MD*MD; j += i) isPrime[j] = false;
memset(dp, 0xff, sizeof(dp));
int t; cin >> t;
while (t--) {
s64 a, b; cin >> a >> b;
cout << (go(b)-go(a-1)) << endl;
}
return 0;
}
In Java :
import java.io.*;
import java.util.Arrays;
/**
* @author Sergey Vorobyev (svorobyev@spb.com)
*/
public class Solution {
MyTokenizer in;
PrintWriter out;
public static void main(String[] args) throws Exception {
Solution instance = new Solution();
instance.go();
}
static class MyTokenizer {
private BufferedReader in;
private String[] buffer = new String[]{};
private int uk;
MyTokenizer() {
in = new BufferedReader(new InputStreamReader(System.in));
}
private String nextToken() throws IOException {
while (true) {
if (uk < buffer.length && !buffer[uk].isEmpty()) {
return buffer[uk++];
}
if (uk >= buffer.length) {
uk = 0;
buffer = in.readLine().split(" ");
} else if (buffer[uk].isEmpty()) {
uk++;
}
}
}
int ni() throws IOException {
return Integer.parseInt(nextToken());
}
long nl() throws IOException {
return Long.parseLong(nextToken());
}
}
private int ni() throws IOException {
return in.ni();
}
private long nl() throws IOException {
return in.nl();
}
boolean[] isPrime;
long[] fact;
long[][][] d;
long[][][] d2;
long[][][][] bufferedD;
int[] primes;
int[] nextPrime;
int primesCount;
static final int MAX_DIGITS = 18;
static final int MAX_SQ_SUM = 81 * MAX_DIGITS + 1;
static final int MAX_PRIME = 3000;
static final int MAX_SUM = 9 * MAX_DIGITS + 1;
void fillPrime() {
primes = new int[MAX_PRIME];
nextPrime = new int[MAX_PRIME];
isPrime = new boolean[MAX_PRIME];
Arrays.fill(isPrime, true);
isPrime[0] = isPrime[1] = false;
int predPrime = -1;
for (int i = 2; i < MAX_PRIME; i++) {
if (isPrime[i]) {
for (int ii = predPrime + 1; ii <= i; ii++) {
nextPrime[ii] = primesCount;
}
predPrime = i;
primes[primesCount++] = i;
for (int j = i * i; j < MAX_PRIME; j += i) {
isPrime[j] = false;
}
}
}
}
void fillD2() {
d2 = new long[MAX_DIGITS + 1][MAX_SUM][MAX_SQ_SUM];
for (int dig = 0; dig <= MAX_DIGITS; dig++) {
for (int sum = 0; sum + dig * 9 < MAX_SUM; sum++) {
for (int isum = 0; isum <= Math.min(9*dig, MAX_SUM - sum); isum++) {
if (isPrime[sum + isum]) {
for (int sqSum = 0; sqSum <= dig*81; sqSum++) {
d2[dig][sum][sqSum] += d[dig][isum][sqSum];
}
}
}
}
}
}
private void go() throws Exception {
long time = System.currentTimeMillis();
in = new MyTokenizer();
out = new PrintWriter(System.out);
fillPrime();
int FAC_MAX = MAX_DIGITS + 1;
fact = new long[FAC_MAX];
fact[0] = 1;
for (int i = 1; i < FAC_MAX; i++) {
fact[i] = fact[i - 1] * i;
}
d = new long[MAX_DIGITS + 1][MAX_SUM][MAX_SQ_SUM];
int[] counts = new int[10];
rec(0 /*pos*/, MAX_DIGITS, 0, 0, 1, true);
fillD2();
int T = ni();
for (int cas = 1; cas <= T; cas++) {
long A = nl();
long B = nl();
out.println(getLucky(B) - getLucky(A - 1));
}
out.close();
}
private long getLucky(long x) {
if (x == 1000000000000000000L) {
return getLucky(x - 1);
}
int n = log10(x);
int[] digits = toDigits(x, n);
return calc(digits);
}
private void rec(int pos, int remind, int sum, int sqSum, long div, boolean update) {
int used = MAX_DIGITS - remind;
if (update) {
d[used][sum][sqSum] += fact[used] / div;
}
if (pos == 10) {
return;
}
int pos2 = pos * pos;
for (int i = 0; i <= remind; i++) {
rec(pos + 1, remind - i, sum, sqSum, div * fact[i], i != 0);
sum += pos;
sqSum += pos2;
}
}
private int[] toDigits(long x, int n) {
int[] res = new int[n];
for (int i = 0; i < n; i++) {
res[n - i - 1] = (int) (x % 10);
x /= 10;
}
return res;
}
private long calc(int[] digits) {
int n = digits.length;
long res = 0;
int sum = 0;
int sumSq = 0;
for (int i = 0; i < n; i++) {
int nmi = n - 1 - i;
int maxprime = 81 * n;
if (digits[i] < 5) {
for (int j = 0; j < digits[i]; j++) {
sum += j;
sumSq += j * j;
for (int p2i = nextPrime[sumSq]; primes[p2i] < maxprime; p2i++) {
int isumSq = primes[p2i] - sumSq;
res += d2[nmi][sum][isumSq];
}
sum -= j;
sumSq -= j * j;
}
} else {
for (int p2i = nextPrime[sumSq]; primes[p2i] < maxprime; p2i++) {
int isumSq = primes[p2i] - sumSq;
res += d2[n-i][sum][isumSq];
}
for (int j = digits[i]; j < 10; j++) {
sum += j;
sumSq += j * j;
for (int p2i = nextPrime[sumSq]; primes[p2i] < maxprime; p2i++) {
int isumSq = primes[p2i] - sumSq;
res -= d2[nmi][sum][isumSq];
}
sum -= j;
sumSq -= j * j;
}
}
sum += digits[i];
sumSq += digits[i] * digits[i];
}
// inclusive
res += isPrime[sum] && isPrime[sumSq] ? 1 : 0;
return res;
}
private int log10(long x) {
int res = 0;
while (x != 0) {
res++;
x /= 10;
}
return res;
}
}
In C :
#include <stdio.h>
#include <math.h>
#include <string.h>
#include <malloc.h>
#include <assert.h>
#define NUMDIGITS 19
#define NUMSIEVE ((NUMDIGITS-1)*9*9)
// Sieve for generating primes
int p[1+NUMSIEVE];
// Lookup table for factorials
unsigned long long factorial[NUMDIGITS+1];
// To store binned integers
int bin[10];
// Stuff for number lists
typedef struct {
short s;
short q;
unsigned long long mult;
} Element;
static Element *lists[NUMDIGITS+1];
unsigned int listlen[NUMDIGITS+1];
unsigned int listidx[NUMDIGITS+1];
unsigned long long table[1+(NUMDIGITS-2)*9][1+(NUMDIGITS-2)*9*9];
static unsigned long long lookup[NUMDIGITS-1][1+(NUMDIGITS-2)*9][1+(NUMDIGITS-2)*9*9];
void make_list_1(int digit, int n, unsigned long long num, short s, short q, int N)
{
int i;
Element *e = lists[n] + listidx[n]++;
e->s = s;
e->q = q;
e->mult = factorial[n]/num;
if (n==N) return;
for (i=digit; i<=9; i++) {
bin[i]++;
make_list_1(i, n+1, num*bin[i], s+i, q+i*i, N);
bin[i]--;
}
}
void pack_list(int n)
{
int i, j, k, idx;
int imax=n*9, jmax=n*9*9, kmax=listlen[n];
Element *e;
for (i=0; i<=imax; i++)
for (j=0; j<=jmax; j++)
table[i][j] = 0;
for (k=0; k<kmax; k++) {
e = lists[n]+k;
table[e->s][e->q] += e->mult;
}
idx = 0;
for (i=0; i<=imax; i++) {
for (j=0; j<=jmax; j++) {
if (table[i][j]) {
e = lists[n]+idx++;
e->s = i;
e->q = j;
e->mult = table[i][j];
}
}
}
listlen[n] = idx;
}
void make_list(int N)
{
int i;
make_list_1(0, 0, 1, 0, 0, N);
for (i=1; i<=N; i++) {
pack_list(i);
}
}
unsigned long long num_lucky(int n, int ss, int sq)
{
int i, imax=listlen[n];
unsigned long long ires = 0;
if (lookup[n][ss][sq]) return lookup[n][ss][sq];
for (i=0; i<imax; i++) {
if (p[sq+lists[n][i].q]&&p[ss+lists[n][i].s]) {
ires += lists[n][i].mult;
}
}
if (ires)
lookup[n][ss][sq] = ires;
return ires;
}
int main()
{
int T;
int i, j;
int nsieve, sqrtnsieve;
int len, alen, blen;
char A[NUMDIGITS+1], B[NUMDIGITS+1];
int a[NUMDIGITS+1], b[NUMDIGITS+1];
unsigned long long ires;
int ssl, ssu, sql, squ;
// Lengths of lists of candidates
for (i=0; i<=NUMDIGITS; i++) {
if (i>0)
listlen[i] = ((i+9)*listlen[i-1])/i;
else
listlen[i] = 1;
lists[i] = (Element *)malloc(listlen[i]*sizeof(Element));
listidx[i] = 0;
}
// Make primes
nsieve = NUMSIEVE;
sqrtnsieve = sqrt(nsieve);
p[1] = 0;
for (i=2; i<=nsieve; i++)
p[i] = 1;
for (i=2; i<=sqrtnsieve; i++)
if (p[i]==1)
for (j=i+i; j<=nsieve; j+=i)
p[j] = 0;
// Make factorials
factorial[0] = 1;
for (i=1; i<=NUMDIGITS; i++)
factorial[i] = i*factorial[i-1];
// Create the master lists
make_list(17);
scanf("%d", &T);
while (T--) {
scanf("%s %s", A, B);
alen = strlen(A);
blen = strlen(B);
if (alen==19) {
alen = 18;
for (i=0; i<alen; i++)
A[i] = '9';
}
if (blen==19) {
blen = 18;
for (i=0; i<blen; i++)
B[i] = '9';
}
len = blen;
for (i=0; i<len; i++)
a[i] = (i<blen-alen)?0:(A[i-(blen-alen)]-'0');
for (i=0; i<len; i++)
b[i] = B[i]-'0';
ires = 0;
for(i=a[0]; i<=b[0]; i++)
ires += num_lucky(len-1, i, i*i);
ssl = sql = ssu = squ = 0;
for (j=1; j<len; j++) {
ssl += a[j-1];
sql += a[j-1]*a[j-1];
for (i=0;i<a[j];i++) {
ires -= num_lucky(len-j-1, ssl+i, sql+i*i);
}
ssu += b[j-1];
squ += b[j-1]*b[j-1];
for (i=9;i>b[j];i--) {
ires -= num_lucky(len-j-1, ssu+i, squ+i*i);
}
}
printf("%llu\n", ires);
}
for (i=0; i<=NUMDIGITS; i++)
if (lists[i]) free(lists[i]);
return 0;
}
In Python3 :
def lucky(N):
p = [int(x) for x in str(N)]
numDigits = len(p)
total = 0
curSum = 0
curSumSq = 0
while len(p):
for a in range(p[0]):
total += resolve(numDigits - 1, curSum + a, curSumSq + a**2)
numDigits -= 1
curSum += p[0]
curSumSq += p[0]**2
p.pop(0)
return total
def resolve(n, s, sq):
if (n,s,sq) in memo:
return memo[(n,s,sq)]
if n == 0:
if s in primes and sq in primes:
memo[(n,s,sq)] = 1
return 1
memo[(n,s,sq)] = 0
return 0
c = 0
for b in range(10):
c += resolve(n-1, s + b, sq + b**2)
memo[(n,s,sq)] = c
return c
memo = {}
primes = set([2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999])
for test in range(int(input())):
[A,B] = [int(x) for x in input().split(' ')]
print(lucky(B+1)-lucky(A))
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