Let's start with 2 and 3 maltiples blanking from set of natural numbers:
M2={n: n=2k}, M3={n: n=3k}, then
N\(M2UM3)={Rn(2;3):Rn(2;3)=3n+(7(-1)n-1-3)/2,n=1,2,...}.
Rn(2;3)= 3n+(7(-1)n-1-3)/2;
Rn(2;3)=3n+1+((-1)n-1+1)/2.
Generalization:
p1=2, p2=3, p3=5, p4=7,...,pп(p)=p,...;
p!=2*3*5*7*11*...*p;
w(p)=п(p!)-п(p)+1.
M2={2k: k=1,2,3,...};
M3={3k: k=1,2,3,...};
M5={5k: k=1,2,3,...};
..............................
Mp={pk: k=1,2,3,...}.
N\(M2UM3UM5U...UMp)={Rn(2,3,5,7,...,p): n=0,1,2,3,...}.
Rn(2,3,5,7,...,p)=p![n/w(p)]+1 when n=0(mod w(p)),
Rn(2,3,5,7,...,p)=p![n/w(p)]+pп(p)+1 when n=1(mod w(p)),
Rn(2,3,5,7,...,p)=p![n/w(p)]+pп(p)+2 when n=2(mod w(p)),
.....................................................................
Rn(2,3,5,7,...,p)=p![n/w(p)]+pw(p)-1 when n=w(p)-1(mod w(p)).
Rn(2,3,5,7,...,p)=p!n/w(p)+1, when w(p)|n;
Rn(2,3,5,7,...,p)=p![n/w(p)]+pi(n), i(n)<=п(p!), when w(p)|n;
w(p)=п(p!)-п(p)+1;
i(n)=п(p)+n-[n/w(p)]w(p).
If rad(n)=p1p2...pr then card N\(Mp1UMp2U...Mpr)=ф(n);
If rad(n)=p!, then п(n)-п(p)<ф(n);
п(p!)-п(p)<ф(p!).
Rф(p!)-1(2,3,5,7,...,p)=p! -1;
Rф(p!)(2,3,5,7,...,p)=p! +1.
rad(ab)=rad(a)rad(b)/rad(a,b);
ф(a)=a*ф(rad(a))/rad(a);
ф(ab)=ф(a)ф(b)ф(rad(b)/rad(a,b))rad(a,b)/ф(rad(b)).
emzari papava
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