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It is possible that I'm wrong, but I think there's an error in an example given in Problem 699. The example in question is T(100)=270.
The sigma function example is good, and a naive implementation of summing the factors of a positive integer will show that σ(10)=1+2+5+10=18.
The naive implementation to which I am referring is one where the square root (if it is an integer) of n is added twice. This incorrect implementation allows T(100)=270, but correcting the function to only count the square root once (if the square root is an integer) makes T(100)=258.
I specifically tested my factorization of 100, and when I was getting T(100)=270, I was counting the factor 10 twice. When I changed sigma to only count the square root once for general n, I got T(100)=258.
Am I just missing an important detail, or is my assumption correct that the square root should only be counted once?

EDIT: I also tested T(10^{6}). I got 26089275 instead of 26089287 using what I think is the correct sigma function, based on the definition given in the problem.

The answer for 10^6 is also correct.
There must be another error in your program.
(it is correct that the divisor 10 of 100 should only be added once.)