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a_n = sum of a_floor(n/k) + 1

Source: 2012 Indonesia Round 2.5 TST 2 Problem 4

May 21, 2012

floor functionnumber theory unsolvednumber theory

Problem Statement

The sequence

a_i

is defined as

a_1 = 1

and

a_n = a_{\left\lfloor \dfrac{n}{2} \right\rfloor} + a_{\left\lfloor \dfrac{n}{3} \right\rfloor} + a_{\left\lfloor \dfrac{n}{4} \right\rfloor} + \cdots + a_{\left\lfloor \dfrac{n}{n} \right\rfloor} + 1

for every positive integer

n > 1

. Prove that there are infinitely many values of

n

such that

a_n \equiv n \mod 2012

.

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