632 Perfect numbers
Appendix: Three important number theoretic functions
Letnbe a positive integer with prime factorization
n=
∏k
i=1
paii
There are several important numbers associated withn.
1.The number of divisors ofnis
d(n)=
∏k
i=1
(1 +ai).
2.The sum of divisors ofn, including 1 andnitself, is
σ(n)=
∏k
i=1
paii+1
pi− 1
.
3.The number of positive integers<nwhich are relatively prime to
nis given by
φ(n)=n
∏k
i=1
(
1 −
1
pi
)
.
These functions are all multiplicative in the sense that
f(mn)=f(m)f(n) whenever gcd(m, n)=1.