## SquareFree semiprimes n, σ(n), φ(n), and σ(n) * φ(n)

$\sigma (n)$   is the sum of divisors function

Euler’s totient function $\phi (n)$   is the number of positive integers not exceeding   $n$   that have no common divisors with   $n$   (other than the common divisor 1).

In other words,   $\phi (n)$   is the number of integers   $m$   coprime to   $n$   such that   $1 \; \leq \; m \; \leq \; n$

Product   =   $\phi (n) \cdot \sigma (n)$

Squarefree semiprimes are numbers   $n$   such that   $\phi \,(n) \; + \; \sigma \,(n) \; = \; 2 \, (n+1)$

These results show a pattern.

Prove that a positive integer   $n$   is the product of two primes differing by 2 iff

$\sigma (n) \cdot \phi (n) \; = \; (n + 1) \,(n - 3)$

Also, prove that a positive integer   $n$   is the product of two primes differing by 4 iff

$\sigma (n) \cdot \phi (n) \; = \; (n + 3) \,(n - 5)$

and a positive integer   $n$   is the product of two primes differing by 6 iff

$\sigma (n) \cdot \phi (n) \; = \; (n + 5) \,(n - 7)$

and a positive integer   $n$   is the product of two primes differing by 8 iff

$\sigma (n) \cdot \phi (n) \; = \; (n + 7) \,(n - 9)$

and a positive integer   $n$   is the product of two primes differing by 10 iff

$\sigma (n) \cdot \phi (n) \; = \; (n + 9) \,(n - 11)$

and a positive integer   $n$   is the product of two primes differing by 12 iff

$\sigma (n) \cdot \phi (n) \; = \; (n + 11) \,(n - 13)$