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Iterating the sum-of-divisors function. (English) Zbl 0866.11003

Let \(\sigma\) be the sum-of-divisors function and define \(\sigma^m(n)= \sigma(\sigma^{m-1}(n))\) for \(m>1\). A number \(n\) is said to be \((m,k)\)-perfect if \(\sigma^m(n)=kn\), so that the classical perfect numbers are \((1,2)\)-perfect. The authors give a table of all \((2,k)\)-perfect numbers up to \(10^9\); for example \(2^5\cdot 3^2\cdot 5\cdot 7^2\cdot 11\cdot 13\cdot 31\) is \((2,19)\)-perfect.
Besides the notorious problem of whether there are infinitely many perfect numbers, one may now ask whether every number is \((m,k)\)-perfect for some suitable values of \(m\) and \(k\). The authors have verified that this is so for \(n\leq 1000\), and some results for \(n\leq 400\) are given. For example, \(n=389\) is \((m,k)\)-perfect with the least value \(m=296\) and \(k= 2^{93}\cdot 3^{10}\cdots\approx 5\cdot 10^{232}\). There are 14 values for \(n\leq 400\) whose corresponding least values for \(m\) exceed those for \(n\), namely \(n=3,11,29,53, 58,59,67, 101,109,131, 149,173, 202,239\). Naturally, the computations for such cases are relatively more complicated.
It had been suggested that, corresponding to any \(n_1,n_2\), there are \(m_1,m_2\) such that \(\sigma^{m_1}(n_1)= \sigma^{m_2}(n_2)\). However, from their computational evidence, the authors believe that this may be false.

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11Y70 Values of arithmetic functions; tables
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Online Encyclopedia of Integer Sequences:

Megaperfect numbers: numbers n where A019294(n) = min {m: n divides sigma^(m) (n)} increases to a record; sigma^(m) means apply the sum-of-divisors function m times.
Records in A019294, number of iterations of the sigma function to reach a multiple of the starting value.
Numbers n such that sigma(sigma(n)) = k*n for some k.
Superperfect numbers: numbers k such that sigma(sigma(k)) = 2*k where sigma is the sum-of-divisors function (A000203).
Let sigma_m(n) be result of applying the sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m(n) = k*n; sequence gives log_2 of the (2,2)-perfect numbers.
Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,3)-perfect numbers.
Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,4)-perfect numbers.
Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,6)-perfect numbers.
Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,7)-perfect numbers.
Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,8)-perfect numbers.
Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,9)-perfect numbers.
Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,10)-perfect numbers.
Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,11)-perfect numbers.
Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,12)-perfect numbers.
Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,13)-perfect numbers.
Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,14)-perfect numbers.
Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (3,k)-perfect numbers.
Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (4,k)-perfect numbers.
Number (> 0) of iterations of sigma (A000203) required to reach a multiple of n when starting with n.
a(n) = sigma(sigma(...(sigma(n))...)) / n, where sigma (A000203) is iterated until a multiple of n is reached.
Even superperfect numbers: 2^(p-1) where 2^p-1 is a Mersenne prime (A000668).
Numbers n such that (A000203(n)+28)/n is an integer.
Last of consecutive coprime iterations of sum-of-divisors function
Count of consecutive coprime iterations of sum-of-divisors function
Number of disjoint trees that appear while iterating the sum of divisors function up to n.
Numbers k for which k = sigma(sigma(x)) = sigma(sigma(y)) for some x and y such that k = x + y.
Number of integers x such that the repeated application of sigma(x)->x leads to n.
a(n) is the number of iterations of the map x->sigma(x) when starting from n before arriving at a number with more than one ancestor, with a(1)=0 and where sigma is the sum of divisors.
Repeatedly applying the map x -> sigma(x) partitions the natural numbers into a number of disjoint trees; sequence gives the (conjectural) list of minimal representatives of these trees.
Number of terms in the sigma(x) -> x subtree whose root is n.
Minimum term in the sigma(x) -> x subtree whose root is n.

References:

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