Slightly Excessive Numbers
No slightly excessive numbers exist (a slightly excessive number, n, is one for which sigma(n) would = 2n+1)
Proof:
Suppose sigma(n) = 2n+1
sigma(pn) == 0 [mod 2] [hcf (p,n)=1]
Let p->99999... [Euclid]
sigma(pn)/p = 2n+1
But L.H.S. == 0 [mod 2] , while R.H.S. == 1 [mod 2]