\[m, \; n\]
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 \[a \gt 1\]
.