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sequence of 2004 integers [n+k+\sqrt{n+k}]

Source: Argentina 2004 OMA L3 p1

May 12, 2024
number theoryalgebra

Problem Statement

For each positive integer nn we consider the sequence of 20042004 integers[n+n],[n+1+n+1],[n+2+n+2],…,[n+2003+n+2003]\left [n+\sqrt{n}\right ],\left [n+1+\sqrt{n+1}\right ],\left [n+2+\sqrt{n+2}\right ],\ldots ,\left [n+2003+\sqrt{n+2003}\right ]Determine the smallest integer nn such that the 20042004 numbers in the sequence are 20042004 consecutive integers.
Clarification: The brackets indicate the integer part.
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