已知数列{an}(n∈N*)满足:an=logn+1(n+2)(n∈N*),定义使a1·a2·a3·……ak为整数的数k

an=log(n+1)(n+2)=lg(n+2)/lg(n+1)
a1a2。。。ak=(lg3/lg2)(lg4/lg3)。。。[lg(k+2)/lg(k+1)]=lg(k+2)/lg2=log2 (k+2)
k+2=2^p
k=2^p -2
令1≤2^p -2≤2004
3≤2^p≤2006
2≤p≤10
M=(2²-2)+(2³-2)+。。。+(2^10 -2)
=(2²+2³+。。。+2^10) -2×9
=4×(2^9 -1)/(2-1) -18
=2026