Multiplicative Functions
Number Theory
Name: _______________________
1) Calculate τ(n) and σ(n) for n = 143, 144, and 145.
2) Find three numbers with τ(n) = 24. Find two numbers with σ(n) = 432.
3) Define σk(n) to be the sum of the kth powers of the divisors of n. For example, σ3(12) = 13 +
23 + 33 + 43 + 63 + 123. Show that σk is multiplicative for any k.
4) A number is called superperfect if σ(σ(n)) = 2n. For example, σ(σ(16)) = σ(31) = 32. Show
that if n = 2a where 2a+1 – 1 is prime then n is superperfect.
5) Consider the function a(n), the product of all positive divisors of n. Prove or disprove that
a(n) is multiplicative.
6) Determine and prove a criterion that decised which numbers have τ(n) odd and which have
τ(n) even. Repeat for σ(n).
A-level) Referring to question 4), prove that an even number is superperfect only if it is of the
given form. (Attach on a separate sheet.)