Linear multiplicative function sieve

concept

Domain is a positive integer function is a function of number theory.

Provided p, q coprime , f [P * Q] = f [P] * f [Q] , f is called a multiplicative function. The divisor, Euler, Mobius.

Any of P, Q, f [P * Q] = f [P] * f [Q] , called multiplicative function f is complete. As one function, i.e., f [i] = 1.

We generally use linear sieve can be pre-multiplicative function.

---

Euler phi

void phii()
{
    phi[1]=1;
    for(int i=2;i<=maxn;i++)
    {
        if(!not_prime[i]) 
        {
            prime[++tot]=i;
            phi[i]=i-1;
        }
        for(int j=1;j<=tot&&prime[j]*i<=maxn;j++)
        {
            not_prime[i*prime[j]]=1;
            if(i%prime[j]==0)
            { 
                Non [i * prime [j]] = non [i] * prime [j];
                break ; 
            } 
            Else non [i * prime [j]] = non [i] * non [prime [j]]; 
        } 
    } 
}

Euler

---

Mobius mu

void init()
{
    notp[1]=1;mu[1]=1;
    for(int i=2;i<=maxn;++i)
    {
        if(!notp[i])prime[++tot]=i,mu[i]=-1;
        for(LL j=1;j<=tot&&i*prime[j]<=maxn;++j)
        {
            notp[i*prime[j]]=1;
            if(i%prime[j]==0)
            {
                mu[i*prime[j]]=0;//Comprising a square- 
                BREAK ; 
            } 
            the else MU [Prime * I [J]] = - MU [I]; // number of prime ++ 
        } 
    } 
}

Mobius

---

Divisor

Submultiple there are two, one is seeking the number of divisors, one is seeking divisors

(Do not want their hands to fight the cause) is taken from this blog

Do not play when 1 forgot

void the init () 
{ 
    NOTP [ . 1 ] = . 1 ; YS [ . 1 ] = . 1 ; dd [ . 1 ] = . 1 ;
     // number divisor ys, dd minimum number of prime factors 
    for (Re int I = 2 ; I <= 50000 ; ++ I) 
    { 
        IF (NOTP [I]) Prime [TOT ++] = I, YS [I] =! 2 , dd [I] = . 1 ;
         for (LL J = Re . 1 ; J < Prime && * I TOT = [J] <= 50000 ; ++ J) 
        { 
            NOTP [I * Prime [J]] = . 1 ;
            IF (I% Prime [J] == 0 ) 
            { 
                YS [I * Prime [J]] YS = [I] / (dd [I] + . 1 ) * (dd [I] + 2 ); 
                dd [I * prime [J]] = dd [I] + . 1 ;
                 BREAK ; 
            } 
            the else 
            { 
                YS [I * prime [J]] YS = [I] * 2 ; // (multiplicative function of the nature, the number of prime divisor of the number of 2 =) 
                dd [i * prime [j]] = . 1 ; // prime [J] is the i and i * prime [j] is the smallest prime factor 
            } 
        } 
    } 
}

The number of divisor

void the init () 
{ 
    NOTP [ . 1 ] = . 1 ; sd [ . 1 ] = . 1 ; sp [ . 1 ] = . 1 ;
     // about sd and number, the polynomial sp smallest prime factors (which may be understood as composed of the minimum quality factor divisor number and the bar) 
    for (Re int I = 2 ; I <= 50000 ; ++ I) 
    { 
        IF ! (NOTP [I]) Prime [TOT ++] = I, SD [I] = I + . 1 , SP = I + . 1 ;
         for (LL J = Re . 1 ; J <= TOT Prime && * I [J] <= 50000 ; ++ J) 
        { 
            NOTP [I * Prime [J]] = . 1;
            if(i%prime[j]==0)
            {
                sd[i*prime[j]]=sd[i]/sp[i]*(sp[i]*prime[j]+1);
                sp[i*prime[j]]=sp[i]*prime[j]+1;
                break;
            }
            else
            {
                sd[i*prime[j]]=sd[i]*sd[prime[j]];
                sp[i*prime[j]]=prime[j]+1;
            }
        }
    }
}

About the number and