On common values of $\phi(n)$ and $\sigma(m)$, II
Kevin Ford, Paul Pollack
TL;DR
This paper investigates the intersection of the ranges of Euler's totient function and the sum-of-divisors function, showing that most values of one are not attained by the other, highlighting their distinct behaviors.
Contribution
It provides new results demonstrating that the sets of values of phi(n) and sigma(n) largely do not overlap, advancing understanding of their ranges.
Findings
Most phi-values are not sigma-values.
Most sigma-values are not phi-values.
Limited intersection between the ranges of phi and sigma.
Abstract
Let phi(n) be Euler's totient function and let sigma(n) be the sum of the positive divisors of n. We show that most phi-values (integers in the range of phi) are not sigma-values and vice versa.
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Taxonomy
TopicsAdvanced Mathematical Identities · History and Theory of Mathematics · Analytic Number Theory Research