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3-adic divisibility of a divisor sum

Source: Bundeswettbewerb Mathematik 2022, Round 1 - Problem 4

March 30, 2022

number theorynumber theory proposeddivisorsum of divisors

Problem Statement

For each positive integer

k

let

a_k

be the largest divisor of

k

which is not divisible by

3

. Let

s_n=a_1+a_2+\dots+a_n

. Show that: (a) The number

s_n

is divisible by

3

iff the number of ones in the ternary expansion of

n

is divisible by

3

. (b) There are infinitely many

n

for which

s_n

is divisible by

3^3