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)
 
 

I’ll start with the Squarefree semiprimes.

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

SQUAREFREE A1

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)

SQUAREFREE 2

 

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)

SQUAREFREE 3

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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