Let $a_i$ represent the numbers of coins pirate $i$ has initially and arranged them from the smallest to the biggest. a)The solutions are $n={7,41,49}$. From $a_1+a_2+...+a_n=2009$ and $a_i=a_1+i-1$ we get, $n(2a_1+n-1)=4018$, which by checking gives the above solutions. b)The answer is $1996$. Note that by doing this operation the numbers $a_1,a_2,a_3,a_4,a_5,a_6,a_7$ are a permutatiton of the numbers $0,1,2,3,4,5,6$ in $(mod$ $7)$. So our goal is to turn the numbers into $0,1,2,3,4,5,1996$. This can easily be done by simply choosing the pirates in the order $a_6$, then $a_5$, then $a_4$... because after $6$ turns we will have the six numbers besides $a_7$ turn into $a_i-1$.

0+1+2+3+4+5+6+1996 is not 2009, I think it should be 1994