MathDB

2

Part of 2012 Serbia Team Selection Test

Problems(1)

f(n)={m : m<=n, sigma(m) is odd}, f(n)|n, for inf. many n

Source: Serbia Additional TST 2012, Problem 2

5/19/2012
Let σ(x)\sigma(x) denote the sum of divisors of natural number xx, including 11 and xx. For every n∈Nn\in \mathbb{N} define f(n)f(n) as number of natural numbers m,m≤nm, m\leq n, for which σ(m)\sigma(m) is odd number. Prove that there are infinitely many natural numbers nn, such that f(n)∣nf(n)|n.
floor functionlimitinequalitiesnumber theory proposednumber theory