Repulsion principle, Euler function, phi

The principle of tolerance and repulsion:

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Can also be expressed as set S as a finite set, then the A∪B formula for two sets: A+b-a∩b (∩: Coincident parts) Three sets of the tolerance formula: A∪b∪c = A+b+c-a∩b-b∩c-c∩a +a∩b∩c Detailed reasoning is as follows: 1, equation right modification = {[ (A+B-A∩B) +c-b∩c]-C∩a}+ A∩b∩c2, the Venturi block tag as shown on the right: 1245 constitutes a,2356 composition b,4567 constitute C3, the right of the equation refers to the 1+2+3+4+5+6 six part: Then A∪b∪c is missing Part 7. 4, the equation to the right [] number +c (4+5+6+7), the equivalent of A∪b∪c added 4+5+6 three parts, minus b∩c (that is, 5+6 two parts), but also add some 4. 5, the right side of the equation {} minus c∩a (that is, 4+5 two parts), A∪b∪c again minus Part 5, then add A∩b∩c (that is 5) is just a∪b∪c. Euler function definition The Euler function Phi (n) represents the number of positive integers (including 1) that are smaller than N and coprime with N. For example: PHI (1) = 1; PHI (2) = 1; PHI (3) = 2; PHI (4) = 2; ... PHI (9) = 6; ...

The general formula and its proof the Euler function to calculate a positive integer n is as follows:
1. Multiply n by the number of primes: n = p1 ^ K1 * p2 ^ k2 * ... * pn ^ kn (here p1, p2, ..., PN is prime) 2. PHI (n) =(p1 ^ k1-p1 ^ (k1-1))*(p2 ^ k2-p2 ^ (k2-1))* ... *(pn ^ KN-PN ^ (kn-1))
=n* (p1-1) (p2-1) ... (pi-1)/(P1*P2*......PI);
=n* (1-1/P1) * (1-1/P2) .... (1-1/PN) However, another method for solving the Euler Phi function value is given on purple potato (the method is very similar to the Sieve method for prime number).

voidPhi_table (intNint*Phi) {     for(intI=2; i<=n;i++) phi[i]=0; Phi[i]=1;  for(intI=2; i<=n;i++)if(!Phi[i]) {         for(intj=i;j<=n;j+=i) {if(!phi[j]) phi[j]=J; PHI[J]=phi[j]/i* (I-1); }    }}

However, I did not understand its principle =

Repulsion principle, Euler function, phi