Don't get Scared By Large Numbers

The number can be rewrtitten as $\dbinom{7}{0}1000^0+\dbinom{7}{1}1000^1+...+\dbinom{7}{7}1000^7=(1000+1)^7=7^7\times11^7\times13^7$ Thus:7+11+13=31

Notice the 'main' numbers:

$1 - 7 -21 - 35 - 35 - 21 - 7 - 1$

These are from pascal's triangle!!

Since there are $2$ zeros in between each pairs of numbers we have:

$1001^7$ as the number in the question.

It is a well known fact that $1001 = 7 \cdot 11\cdot 13$ so we have the required sum as $7+11+13 = 31$