2006 Putnam A3

Let $1, 2, 3, \dots, 2005, 2006, 2007, 2009, 2012, 2016, \dots$

be a sequence defined by $x_k = k$ for $k=1, 2, \dots, 2006$ and

$x_{k+1} = x_k + x_{k-2005}$ for $k \geq 2006$. Show that the sequence has

2005 consecutive terms each divisible by 2006.

设 $1, 2, 3, \dots, 2005, 2006, 2007, 2009, 2012, 2016, \dots$

是由 $x_k = k$ 定义的序列，其中 $k=1, 2, \dots, 2006$ 和

$x_{k+1} = x_k + x_{k-2005}$ 对于 $k \geq 2006$。表明序列有

2005 年连续任期可被 2006 年整除。