A Congruence Relation between τ(n) and σ(n)

Proceeding on the lines of my paper (1) on the "Congruence Properties of τ(n)", it can be shown that for any odd prime p,

σ (P) = P+I (mod 8)

= σ(P) (mod 8),

where τ(n) is Ramanujan's function.