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Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) < 2x$, then $x$ is said to be deficient, while if $\sigma(x) > 2x$, $x$ is said to be abundant. (Of course, when $\sigma(x) = 2x$, $x$ is called a perfect number.)

In Advanced Problem 5967, C. W. Anderson (The American Mathematical Monthly, Vol. 82, No. 10 (Dec., 1975), pp. 1018-1020) computed that the density of deficient odd numbers is at least $$\dfrac{48 - 3{\pi}^2}{32 - {\pi}^2} \approx 0.831.$$

Behrend, Wall et al, Deleglise and subsequently Kobayashi computed ever-sharper bounds for the density $A(2)$ of abundant integers. I give Deleglise's bounds here: $$0.2474 < A(2) < 0.2480.$$

In

Kanold, H. J., Uber die Dichten der Mengen der vollkommenen und der befreundeten Zahlen, Math. Z., vol 61, (1954) pp 180–185,

Kanold showed that the density of perfect numbers is $0$.

Since

Density of all numbers $=$ Density of even deficient numbers $+$ Density of odd deficient numbers $+$ Density of abundant numbers $+$ Density of perfect numbers,

and since

Density of even deficient numbers $\geq 0$,

we have the contradiction $$1 > 0 + 0.831 + 0.2474 + 0 \approx 1.0784.$$

Here is my question:

Does this apparent contradiction imply that there is an ERROR in one of the cited papers above?

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Clearly the density of odd deficient numbers cannot be more than $0.5$ in the sense that you interpreted it. Thus Anderson's result must mean that the density of deficient numbers within the odd numbers is at least $0.831$. This interpretation fits well with what is visible on the first page of the paper that you linked to. This would resolve the contradiction.

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  • $\begingroup$ Thank you very much for your quick answer, joriki! I myself was kinda worried about Anderson's result, not until this answer of yours enlightened me! Again, my profuse thanks. =) $\endgroup$ – Jose Arnaldo Bebita-Dris Apr 6 '16 at 10:49

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