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Source. Section 4, pp. 98-99 (text on p. 98, Table 4 on p. 99), of Graeme L. Cohen and Herman J. J. te Riele, Iterating the Sum-of-Divisors Function, Experimental Mathematics 5 (1996), no. 2, 91-100, as identified on the source card. The paper gives these observations no number.

Statement

Setting (pp. 91-92, 94). Write $\sigma^0(n)=n$ and $\sigma^m(n)=\sigma(\sigma^{m-1}(n))$ for $m\ge1$, and let $\widetilde m(n)$ be the least $m\ge1$ with $n\mid\sigma^m(n)$. The paper quotes, from Erdős, Granville, Pomerance and Spiro (1990) by way of Guy's Unsolved Problems in Number Theory, six statements that those authors could neither prove nor disprove (p. 92), among them

• (ii) for any $n>1$, $\sigma^{m+1}(n)/\sigma^m(n)\to\infty$ as $m\to\infty$;
• (iii) for any $n>1$, $(\sigma^m(n))^{1/m}\to\infty$ as $m\to\infty$.

It says it will give computational evidence that statements (ii), (iii), (iv) and (v) are true and that (i) and (vi) are false (p. 92).

Numerical evidence for (iii) (p. 98). Call $N$ megaperfect when $\widetilde m(n)<\widetilde m(N)$ for all $n<N$; the paper lists those below 1000, the last three being $N=401,461,659$ with $\widetilde m(N)=380,557,1287$. With

$$h(n)=\frac{\bigl(\sigma^{\widetilde m(n)}(n)\bigr)^{1/\widetilde m(n)}}{\log\widetilde m(n)},$$

it reports $h(401)=1.1146$, $h(461)=1.1276$ and $h(659)=1.1658$, and says these suggest that $(\sigma^m(n))^{1/m}$ is at least of the same order as $\log m$ as $m\to\infty$, for any $n$.

Table 4 (p. 99). For each of the 21 values of $n\le199$ listed in (4.2) (p. 97), let $j_1$ and $j_2$ be the least $i$ with $\sigma^i(n)>10^{100}$ and $\sigma^i(n)>10^{200}$ respectively, and put $\alpha_u=\sigma^{j_u+1}(n)/\sigma^{j_u}(n)$ and $\beta_u=(\sigma^{j_u}(n))^{1/j_u}$. In the table, $\beta_1/\log j_1$ lies between 0.98176 and 1.08675, $\beta_2/\log j_2$ between 1.03978 and 1.10411, $\alpha_1$ between 5.9943 and 7.1219, and $\alpha_2$ between 7.1539 and 8.3219. The paper presents the table as its summary of the investigation of statements (i), (ii) and (iii) (p. 98).

Conditional implication (p. 98). If statement (iii) holds and the sequence $\bigl((\sigma^i(n))^{1/i}\bigr)_i$ is eventually monotone, then statement (ii) holds: the paper derives this from the fact that $(\sigma^{i+1}(n))^{1/(i+1)}>(\sigma^i(n))^{1/i}$ implies $\sigma^{i+1}(n)/\sigma^i(n)>(\sigma^i(n))^{1/i}$. It adds that its computations strongly suggest the sequence is eventually monotone for every $n$; it does not prove this.

Proof pointer

The implication is the one-line inequality above (p. 98). The numerical values are computations of the paper; this page has not rerun them.

Dependencies

The values of $\widetilde m(n)$ from the paper's extended computations (p. 98) and the sequences of (4.2) (p. 97); see the tree conjecture for the latter. Read depth: claims checked; the statements, values and implication were read on pp. 92 and 98 and Table 4 on p. 99.

Bears on

• Problem 410: statement (iii) is the problem's question. The paper offers numerical evidence for it and proves nothing about it; the conditional implication runs from (iii) to statement (ii) and does not touch the problem's answer.