Aliquot iteration and generalizations

E. Catalan³⁷⁴ stated empirically that if $n_1$ is the sum of the divisors $<n$ of $n$, and $n_2$ is the sum of the divisors $<n_1$ of $n_1$, etc., then $n,n_1,n_2,\ldots$ have a limit $\lambda$, where $\lambda$ is unity or a perfect number.

J. Perrott³⁷⁵ [Perott] noted that there is no limit for $n=220$, since

$$n_1=n_3=\cdots=284,\qquad n_2=n_4=\cdots=220.$$

P. Poulet³⁸⁷ discovered the chain of period five,

$$n=12496=2^4\cdot11\cdot71,\quad s(n)=2^4\cdot19\cdot47,\quad s^2(n)=2^4\cdot967,\quad s^3(n)=2^3\cdot23\cdot79,\quad s^4(n)=2^3\cdot1783,$$

with $s^5(n)=n$; and noted that 14316 leads a chain of 28 terms.

Daniel Schwenter⁵² noted in 1636 that 27 and 35 have the same sum of aliquot parts. Kraft³⁶³ noted in 1749 that this is true of the pairs $45,3\cdot29$; $39,55$; $93,145$; and $45,13\cdot19$. In 1823, Thomas Taylor¹⁰² called two such numbers imperfectly amicable, citing the pairs 27, 35; 39, 55; 65, 77; 51, 91; 95, 119; 69, 133; 115, 187; 87, 247. George Peacock⁴⁰⁰ used the same term.

L. E. Dickson⁴⁰³ defined an amicable triple to be three numbers such that the sum of the aliquot divisors of each equals the sum of the remaining two numbers. After developing a theory analogous to that by Euler³⁶⁴ for amicable numbers, Dickson obtained eight sets of amicable triples in which two of the numbers are equal, and two triples of distinct numbers:

$$\begin{array}{lll}293\cdot337a,&5\cdot16561a,&99371a\qquad(a=2^5\cdot3\cdot13),\\3\cdot89b,&11\cdot29b,&359b\qquad(b=2^{14}\cdot5\cdot19\cdot31\cdot151).\end{array}$$

Catalan’s empirical claim and Perrott’s objection

Let $s(n)$ be the sum of the proper positive divisors of $n$. Catalan³⁷⁴ empirically claims that the sequence $n,s(n),s^2(n),\ldots$ has a limit, either 1 or a perfect number. Superscripts here mean repeated application, not powers.

Perrott³⁷⁵ [Perott] points out that there is no limit for $n=220$: the odd iterates are 284 and the even iterates are 220.

Poulet’s longer cycles

Poulet³⁸⁷ discovers a cycle of length five:

$$n=12496=2^4\cdot11\cdot71,\quad s(n)=2^4\cdot19\cdot47,\quad s^2(n)=2^4\cdot967,$$ $$s^3(n)=2^3\cdot23\cdot79,\quad s^4(n)=2^3\cdot1783,\quad s^5(n)=n.$$

He also notes that 14316 begins a cycle of 28 terms.

Equal proper-divisor sums

Schwenter⁵² notes in 1636 that $s(27)=s(35)$. Kraft³⁶³ gives the pairs $(45,87)$, $(39,55)$, $(93,145)$ and $(45,247)$ in 1749. In 1823, Taylor¹⁰² calls pairs with the same proper-divisor sum imperfectly amicable, listing $(27,35)$, $(39,55)$, $(65,77)$, $(51,91)$, $(95,119)$, $(69,133)$, $(115,187)$ and $(87,247)$. Peacock⁴⁰⁰ uses the same term.

Amicable triples

Dickson⁴⁰³ defines an amicable triple $(A,B,C)$ by

$$s(A)=B+C,\qquad s(B)=A+C,\qquad s(C)=A+B.$$

He develops a theory analogous to Euler's³⁶⁴ for pairs, and obtains eight triples with two equal entries and two with distinct entries:

$$a(293\cdot337,\ 5\cdot16561,\ 99371),\qquad a=2^5\cdot3\cdot13,$$ $$b(3\cdot89,\ 11\cdot29,\ 359),\qquad b=2^{14}\cdot5\cdot19\cdot31\cdot151.$$

Iterating the proper-divisor sum

Let $s(n)=\sigma(n)-n$ for positive integers $n$. Starting at $n_0$, form $n_{j+1}=s(n_j)$ while $n_j>0$. If 0 is reached, stop: we do not define $s(0)$. A prime goes to 1 and then to 0. Some historical accounts stop at 1 instead.

A perfect number is a fixed point, $s(n)=n$. A distinct amicable pair is a cycle of length two. Once any value repeats, determinism forces the subsequent sequence to repeat. Thus the cycle $220\to284\to220$ already refutes Catalan's claim of convergence to 1 or a perfect number, exactly as Perrott observes on p. 48.

A bounded computation has three possible outcomes: termination, a detected cycle, or an unfinished trajectory. An unfinished trajectory is not proof of divergence or of eventual periodicity. The empirical revision reported by Dickson on p. 49 is still an empirical report within this edition, not a theorem established by the calculator.

Longer cycles

Poulet's five-cycle becomes

$$12496\to14288\to15472\to14536\to14264\to12496.$$

The first step is directly visible from the factorization:

$$\sigma(12496)=\sigma(2^4)\sigma(11)\sigma(71)=31\cdot12\cdot72=26784,$$

so $s(12496)=14288$. Applying the same calculation to each of the next four terms proves the displayed cycle. Its five entries are distinct, so its least period is five. The 28-term cycle starting at 14316 is likewise a finite certificate: all successive sums and the return to the starting value can be checked exactly.

Equal images are a different relation

The condition $s(M)=s(N)$ does not say that $M,N$ form a cycle. For example,

$$s(27)=1+3+9=13,\qquad s(35)=1+5+7=13.$$

Both numbers map to 13; neither maps to the other. Dickson records the names “imperfectly amicable” and, in another notice on p. 50, nombres associés. In the language of a function, these numbers belong to the same fiber: the set of inputs with a specified output. That term labels the relation without changing the calculation.

Conversely, a distinct amicable pair has $s(M)=N\ne M=s(N)$, so its two proper-divisor sums are different. It has equal total divisor sums instead. Confusing $s$ and $\sigma$ would erase the distinction.

Amicable triples are not three-cycles

Dickson's amicable triple satisfies $s(A)=B+C$ and its two cyclic counterparts. Equivalently,

$$\sigma(A)=\sigma(B)=\sigma(C)=A+B+C.$$

A three-cycle would instead require $s(A)=B$, $s(B)=C$, $s(C)=A$. These are different equations.

For Dickson's first distinct triple, put $a=2^5\cdot3\cdot13=1248$ and

$$A=293\cdot337a,\quad B=5\cdot16561a,\quad C=99371a.$$

The three residual divisor sums coincide:

$$294\cdot338=6\cdot16562=99372.$$

Also $293\cdot337+5\cdot16561+99371=280917$, while $\sigma(a)=63\cdot4\cdot14=3528$. Since

$$3528\cdot99372=1248\cdot280917,$$

multiplicativity gives the required common total. The relevant residual factors are prime and coprime to $a$; the accompanying exact-integer checks verify these hypotheses as well as the sum identity. The second triple in Modernized is checked in the same way.

If repeats are allowed, the triple $(N,N,N)$ satisfies the equations exactly when $\sigma(N)=3N$. Thus the generalization reconnects to multiply perfect numbers. More generally, a tuple of $k$ equal entries corresponds to multiplicity $k$. This is an elementary equivalence, not a claim of a new discovery.