[PAST EVENT] Mathematics Colloquium: Jared Lichtman (Dartmouth)
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Mathematics Colloquium: Jared Lichtman (Dartmouth)
Title: The reciprocal sum of primitive nondeficient numbers
Abstract: The ancients were enamored by numbers that were equal to the sum of their own proper divisors, and hailed such numbers as perfect. In modern notation, $\sigma(n)$ denotes the sum of divisors of $n$, so $n$ is perfect if $\sigma(n)=2n$. Then $n$ is called abundant if $\sigma(n)>2n$, and deficient if $\sigma(n)<2n$. Additionally if $n$ is nondeficient, then by multiplicativity of $\sigma(n)/n$ all multiples of $n$ are also nondeficient. Thus, we are led to define primitive nondeficient (pnd) numbers to be nondeficient numbers all of whose proper divisors are deficient.
In 1934, Erd?s showed that the reciprocal sum of pnds converges, which he used to prove that the abundant numbers have a natural density. In this talk, we will outline some of recent work showing the reciprocal sum of pnds is between 0.348 and 0.380.