| 1 | 2 | 3 | 4 | 5 | 6 |
| 0/1 | 0/1 | 1/2 | 1/3 | 2/3 | 1/4 |
| 7 | 8 | 9 | 10 | 11 |
| 3/4 | 1/5 | 2/5 | 3/5 | 4/5 |
1st row = k (1….10^10)
2nd row = Rational number ( 0 to 1)
আলোচনাঃ
প্রশ্ন অনুসারে আমাকে K দেওয়া থাকবে । আমাকে k-তম Rational number বের করা লাগবে।
তার মানে যদি k=5 হয় তাহলে আমার উত্তর হবে 2/3. এখানে n=2 আর d=3 এবং gcd(n,d)=1 হবে। তার মানে Rational number এর হর এবং লব এর gcd=1 হবে। অর্থাৎ এদেরকে কখনো কাটাকাটি করা যাবে না।
আর একটা শর্ত হচ্ছে, n,d থেকে ছোট হবে কারণ ( n = 0 to d) তবে আমাকে rational number দেওয়া নাই । তাহলে মনে আসতে পারে আগে rational number গুলো generate করবো তারপর k-তম Rational number return করে দেব। কিন্তু k=12,158,598,919 = 10^10 ধরতে পারি। কিন্তু k এর মান অনেক বড় তাই এটা করা সম্ভব না। তাহলে কি করা যায়?আমাদের মূল কাজ হচ্ছে হর এবং লব বের
করা। এটা কিভাবে করা যায় সেটা দেখা যাক।
উত্তরঃ
হর বের করাঃ
প্রশ্নে দেওয়া rational number গুলোর দিকে একটু লক্ষ্য করা যাক,
যখন, d=2 তখন rational number 1 টা
যখন, d=3 তখন rational number 2 টা
যখন, d=4 তখন rational number 2 টা
যখন, d=5 তখন rational number 4 টা
এই number গুলো কি একটু চেনাচেনা লাগে?
phi[2]=? 2 না? তেমনিভাবে phi[3]=2, phi[4]=2 ,phi[5]=4.তাহলে k=5 হয় তাহলে আমরা কি বলতে পারি যে, হর (d)=5 তখন rational number(মূলদ সংখ্যা) থাকবে 4 টা? তার মানে এখানে আমাদের Euler phi লাগবে। এবার প্রশ্নের 3 no case এর দিকে একটু তাকাই। যখন k= 12,158,598,919 হলে হর 200000. তাহলে আমাদের 200000 পর্যন্ত Euler phi বের করলেই হবে।
আমরা জানি, phi[1]=1 কিন্তু প্রশ্নে d যখন 1 তখন rational number দুইটা ।তাই phi[1]=2 দিয়ে initialize করতে পারি। তাহলে,
| N= 1 | 2 | 3 | 4 | 5 |
| Phi[N] = 2 | 1 | 2 | 2 | 4 |
| Prefix sum= 2 | 3 | 5 | 7 | 11 |
উপরে prefix sum যখন 2 , এটার মানে হচ্ছে যখন N=1 তখন পর্যন্ত 2 টা rational number পাওয়া গেছে। N যখন 2, prefix sum তখন 3. তার মানে এখন পর্যন্ত 3 টা rational number পাওয়া গেছে। তেমনিভাবে N যখন 4, prefix sum তখন 7. তার মানে এখন পর্যন্ত 7 টা rational number পাওয়া গেছে। আর এই N=4 এ (6 থেকে 7 তম) rational number আছে।
তাহলে k=10 হলে, 10 তম rational number কোথায় থাকবে? N=5 এর মধ্যে থাকা উচিত না? কারণ এই part এ 8 থেকে 11 তম rational number থাকবে। আর এই 8-11 তম rational number এর হর কত হওয়া উচিত? হর কি 5 হওয়া উচিত না? এইটা উপরেই আলোচনা করেছি। যখন, d=5 তখন rational number 4 টা । আর এই 4 টা number হচ্ছে 8-11 তম rational number.তাই এদের হর=5. তাহলে হর পাবার উপায় হচ্ছে, প্রথমে Euler phi calculate করবো। এরপর এদের prefix sum বের করব। এবার ‘k’ এর lower bound করলেই হর পেয়ে যাবো।
লব পাবার উপায়ঃ
হর পেয়ে গেলে লব পাওয়া তেমন কঠিন কোনো কাজ না। k=10 হলে আমাদের উত্তর হবে 3/5. তার মানে লব=3 . এই লব আসলে থাকবে কোথায়? prefix sum
যখন 7 তখন হর=4 হবে ।তার মানে prefix sum = 7 পর্যন্ত আমাদের হরই পাবার আশা নাই । যখন prefix sum = 8 তখন হর=5 . এখন আমরা লব পেতে পারি। এখন
একটা জিনিস খেয়াল করি, k=10 হলে এর আগের prefix sum = 7 ছিল। এখন 10-7=3 ( k-phi[LB-1] ) । এই 3 হচ্ছে 3 no position এ আমাদের কাংক্ষিত লব আছে এটা বুঝায়।
এখন 1-3 পর্যন্ত brute force করলেই আমরা লব পেয়ে যাব।
C++ Code:
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const ll N=2e5+5;
ll phi[N];
void ETFPre(){
phi[0]=0;
phi[1]=2;
for(ll i=2;i<=N;i++) phi[i]=i;
for(ll i=2;i<=N;i++){
if(phi[i]==i){
for(ll j=i;j<=N;j+=i){
phi[j]-= phi[j]/i;
}
}
phi[i]=phi[i]+phi[i-1];
}
}
int main(){
ETFPre();
ll t,n,k;
while(cin>>k && k!=0){
if(k==1){
printf("0/1\n");
continue;
}
if(k==2){
printf("1/1\n");
continue;
}
ll LB=lower_bound(phi,phi+N,k)-phi; // to find denominator(হর)
ll sub=k-phi[LB-1]; // position of numerator (লব)
ll cnt=0;
for(ll i=1;i<=LB;i++){
if(__gcd(i,LB)==1){
cnt++;
if(cnt==sub){
printf("%lld/%lld\n",i,LB);
break;
}
}
}
}
}
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