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

Source: Serbia Additional TST 2012, Problem 2

May 19, 2012
floor functionlimitinequalitiesnumber theory proposednumber theory

Problem Statement

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.
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