Amicable Pairs

A pair of integers  
\[m, \; n\]
  is said to be an amicable pair if  
\[\sigma (m)= \sigma (n)=m+n\]
  where  
\[\sigma (n)\]
  is the sum of the divisors of  
\[n\]
.
220 and 284 are an amicable pair.  
\[\sigma (220)=1+2+4+5+10+11+20+22+44+55+110+220=504\]
  and  
\[\sigma (284)=1+2+4+71+142+284=504\]
  and 220+284=504.
No prime can be one of an amicable pair since  
\[\sigma (p)=p+1\]
  so the condition  
\[\sigma (p)=p+a\]
  cannot hold for any  
\[a \gt 1\]
.