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Statement

Setting (p. 439). $\sigma(n)$ is the sum of the positive divisors of $n$ and $\phi(n)$ is Euler's function. A positive integer $n$ is aliquot when $n=\sigma(m)-m$ for some positive integer $m$, and nonaliquot (untouchable) otherwise; $N_a(x)$ is the set of nonaliquot $n$ with $1\le n\le x$.

Theorem 1 (p. 440). For every positive integer $M$,

$$\lvert N_a(x)\rvert\ \ge\ g_Mx+o_M(x), \qquad g_M=\sum_{d\mid M}\frac{\phi(M/d)}{M/d}\max\Bigl\{0,\ \frac1{2d}-\frac1{\sigma(2d)-2d}\Bigr\}.$$

The numerical value (p. 440). For $M=2^6\times3^5\times5^4\times7^3\times11^2\times13\times17\times19\times23\times29\times31\times37\times41$ the paper states $g_M>0.0602757$, so $\lvert N_a(x)\rvert\ge0.06x+o(x)$. The abstract (p. 439) states the bound for the even numbers below $x$ that are not of the form $\sigma(m)-m$, which is what the proof counts. The earlier bound it improves is Banks and Luca's $\lvert N_a(x)\rvert\ge\frac{x}{48}(1+o(1))$ (p. 440).

The constant $g$ (p. 440). The paper sets $g=\sup g_M$, says that one can prove $g_M<g$ for every positive integer $M$, and conjectures $g<0.07$. It then poses four questions: whether $\lvert N_a(x)\rvert=gx+o(x)$ (Question 1); whether a positive proportion of the even numbers are aliquot (Question 2); an approximate numerical value of $g$ (Question 3); and whether $g$ is irrational (Question 4). None is answered in the paper.

For context the paper notes (p. 439) that almost all odd numbers are aliquot, by the almost-all form of the binary Goldbach problem: if $2n=p+q$ with distinct primes $p,q$ then $2n+1=\sigma(pq)-pq$. Hence $\lvert N_a(x)\rvert\le\frac12x+o(x)$.

Proof pointer

Section 2, pp. 440--442. Only even $2n\le x$ are counted. Representations $2n=\sigma(m)-m$ with $m$ odd cover $o(x)$ such $2n$ by Banks and Luca; for $m$ even, $m\le2x$, and Lemma 1 of the paper (p. 440: for each positive integer $k$, $k\mid\sigma(n)$ for all but $o_k(x)$ of the $n\le x$; p. 441 calls it a weak form of a lemma of De Koninck and Luca) discards the $m$ with $2M\nmid\sigma(m)$. Splitting the even $2n\le x$ by $d=(n,M)$, a representation with $2M\mid\sigma(m)$ forces $(m,2M)=2d$, and $\sigma(m)-m\ge(\sigma(2d)-2d)\,m/(2d)$ bounds the number of such $m$; the class of $d$ then keeps at least the share $\frac{\phi(M/d)}{M/d}\max\{0,\frac1{2d}-\frac1{\sigma(2d)-2d}\}$ of $x$ as nonaliquot, up to $O(\phi(M/d))$, and summing over $d\mid M$ gives the theorem. The numerical value of $g_M$ is stated, not derived, in the paper.

Read depth

Claims checked: the setting, Theorem 1, the numerical value, the remarks on $g$ and the four questions were read clause by clause on the page images of the print, and the proof on pp. 440--442 was followed. The value $g_M>0.0602757$ was not recomputed. Nothing here is independently reviewed.

Dependencies

None in the corpus. External inputs named by the paper: Banks and Luca's bound $o(x)$ for the even numbers $\sigma(m)-m$ with $m$ odd (Colloq. Math. 103 (2005)) and De Koninck and Luca's lemma behind Lemma 1 (Colloq. Math. 108 (2007)).

Source. Y.-G. Chen and Q.-Q. Zhao, Nonaliquot numbers, Publ. Math. Debrecen 78 (2011), no. 2, 439--442, doi:10.5486/PMD.2011.4820; the edition read is named on the source card.

Bears on

• Problem 418: adjacent only. The theorem bounds the integers not of the form $\sigma(m)-m$, the companion of the problem's function $n-\phi(n)$; it says nothing about the values of $n-\phi(n)$.