Sorting the Values of Two Number Theoretic Functions

The function (the Euler phi function of , also known as the totient function) gives the number of positive integers relatively prime to . For example, because 1, 3, 7, 9 are relatively prime to 10. The function is the sum of the divisors of . For example, because the divisors of 10 are 1, 2, 5, 10 and .
The plot shows either the unsorted or sorted values of or for from 1 to up to a maximum of 6000. You can drag the locator to see that the line through the bulk of the unsorted values corresponds to the mostly linear part of the sorted values on the left. For large values the plot of the sorted values has a visible change in curvature on the right.


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