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Source. Section 4, pp. 97-98, 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 the conjecture no number.

Statement

Setting (pp. 91-92, 97). Write $\sigma^0(n)=n$ and $\sigma^m(n)=\sigma(\sigma^{m-1}(n))$ for $m\ge1$. Statement (vi) of the paper's list (p. 92), quoted from Erdős, Granville, Pomerance and Spiro (1990), reads: for any $n_1,n_2>1$ there are $m_1,m_2$ with $\sigma^{m_1}(n_1)=\sigma^{m_2}(n_2)$. The paper says it does not believe statement (vi) is true (p. 97).

Trees (p. 97). The paper calls a set of starting values a $(\pi_1,\pi_2,\pi_3)$-tree when $\pi_1$ is its smallest number and every sequence $(\sigma^i(n))_{i\ge1}$ with $\pi_1\le n\le\pi_2$ in it meets the sequence $(\sigma^i(\pi_1))_{i\ge1}$ while $\sigma^i(n)<\pi_3$. Fixing $\pi_2$ and $\pi_3$ determines the successive trees for all $\pi_1\le\pi_2$.

Computation (pp. 97-98). There are 21 $(\pi_1,200,10^{200})$-trees, with

$$\pi_1\in\{2,5,16,19,27,29,33,49,50,52,66,81,85,105,146,147,163,170,189,197,199\}. \qquad(4.2)$$

The paper computed $(\sigma^i(n))$ for each $n$ with $2\le n\le200$, grouped the sequences by whether the first term above $10^{10}$ occurs in an earlier sequence, which gave 21 $(\pi_1,200,10^{10})$-trees, and then compared the first terms above $10^{200}$: the trees remained distinct. It also found 64 $(\pi_1,1000,10^{100})$-trees.

Conjecture (p. 98). The 21 trees for $\pi_2=200$ remain distinct as $\pi_3\to\infty$; in the paper's words, "we conjecture that this will stay true as $\pi_3\to\infty$".

Related observation (p. 97). Writing $n_1,n_2$ for $n,tn$ in Theorem 3.1, the paper notes that any pair with $n_1\widetilde k(n_1)=n_2\widetilde k(n_2)$ (4.1) gives $\sigma^{m_1}(n_1)=\sigma^{m_2}(n_2)$ with $m_i=\widetilde m(n_i)$, and lists nine such pairs from Table 2 in which $n_2$ is not a multiple of $n_1$: $(7,24)$, $(9,168)$, $(10,12)$, $(14,24)$, $(18,120)$, $(36,168)$, $(62,96)$, $(72,336)$ and $(341,384)$.

Proof pointer

None: the conjecture is supported only by the computation, which this page has not rerun.

Dependencies

The paper's computation of the sequences $(\sigma^i(n))$ for $2\le n\le200$ up to $10^{200}$. Read depth: claims checked; the definitions, the computation report and the conjecture were read on pp. 97-98.

Bears on

• Problem 412: statement (vi) is the problem's question. The conjecture, if true, would give pairs such as $n_1=2$, $n_2=5$, roots of different trees, with $\sigma^{m_1}(n_1)\ne\sigma^{m_2}(n_2)$ for all $m_1,m_2\ge1$, a negative answer. The paper establishes only that the trees do not meet below $10^{200}$, which decides nothing.