Let’s say the divisors of n are d(1), d(2)........d(n) where d(1)<d(2)<....<d(n). The largest proper divisor of n is $d(n-1)$ which is equal to $n/d(2)$ as $d(2).d(n-1)= n$ Therefore, similarly $d(n-2) = n/d(3)$ and $d(n-3)=n/d(4)$ According to the question, $d(n-1)+d(n-2)+d(n-3)= n$ Therefore, $n/d(2)+n/d(3)+n/d(4)= n$ So, $1/d(2) +1/d(3)+1/d(4)=1$ As we know only integer solutions for 1/a+1/b+1/c=1 are (2,3,6) (3,3,3) (4,4,2) And here d(2)<d(3)<d(4) . Therefore, (2,3,6) are the only possible ones. d(2)=2, d(3)=3, d(4)=6 ....Upto 3000 no.of multiples of 6 is 500 out of which 250 are 6. odd (not being multiple of 2) and multiples of 30 (5.6) out of those 250 are 250/5=50. Therefore no. of nice integers not greater than 3000 is 250-50 =200.