Nowhere in mathematics is the progress resulting from the advent of computers as apparent as in the additive number theory. In this part, we describe the role of computers in the investigation of the ...oldest function studied in mathematics, the divisor sum. The disciples of Pythagoras started to systematically explore its behavior more than 2500 years ago. A description of the trajectories of this function—perfect numbers, amicable numbers, sociable numbers, and the like—constitute the contents of several problems stated over 2500 years ago, which still seem completely impenetrable. The theorem of Euclid and Euler reduces classification of
even
perfect numbers to Mersenne primes. After 1914 not a single new Mersenne prime was ever produced by hand, and since 1952 all of them have been discovered by computers. Using computers, now we construct hundreds or thousands times more new amicable pairs
daily
than were constructed by human beings over several millennia. At the end of the paper, we discuss yet another problem posed by Catalan and Dickson.
On the distribution of sociable numbers Kobayashi, Mitsuo; Pollack, Paul; Pomerance, Carl
Journal of number theory,
08/2009, Letnik:
129, Številka:
8
Journal Article
Recenzirano
Odprti dostop
For a positive integer
n, define
s
(
n
)
as the sum of the proper divisors of
n. If
s
(
n
)
>
0
, define
s
2
(
n
)
=
s
(
s
(
n
)
)
, and so on for higher iterates.
Sociable numbers are those
n with
s
...k
(
n
)
=
n
for some
k, the least such
k being the
order of
n. Such numbers have been of interest since antiquity, when order-1 sociables (perfect numbers) and order-2 sociables (amicable numbers) were studied. In this paper we make progress towards the conjecture that the sociable numbers have asymptotic density 0. We show that the number of sociable numbers in
1
,
x
, whose cycle contains at most
k numbers greater than
x, is
o
(
x
)
for each fixed
k. In particular, the number of sociable numbers whose cycle is contained entirely in
1
,
x
is
o
(
x
)
, as is the number of sociable numbers in
1
,
x
with order at most
k. We also prove that but for a set of sociable numbers of asymptotic density 0, all sociable numbers are contained within the set of odd abundant numbers, which has asymptotic density about 1/500.
In this paper we present a new computational record: the aliquot sequence starting at 3630 converges to 1 after reaching a hundred decimal digits. Also, we show the current status of all the aliquot ...sequences starting with a number smaller than 10,000; we have reached at leat 95 digits for all of them. In particu lar, we have reached at least 112 digits for the so-called "Lehmer five sequences," and 101 digits for the "Godwin twelve sequences." Finally, we give a summary showing the number of aliquot sequences of unknown end starting with a number less than or equal 10
6
.
Advances in aliquot sequences Benito, Manuel; Varona, Juan L.
Mathematics of computation,
01/1999, Letnik:
68, Številka:
225
Journal Article
Recenzirano
Odprti dostop
In this paper we describe some advances in the knowledge of the behavior of aliquot sequences starting with a number less than 10000. For some starting values, it is shown for the first time that the ...sequence terminates. The current record for the maximum of a terminating sequence is located in the one starting at~4170; it converges to~1 after~869 iterations getting a maximum of~84 decimal digits at iteration~289.
If the natural number n has the canonical form (The equation is abbreviated) then (The equation is abbreviated) is said to be an exponential divisor of n if b_i|a_i for I = 1, 2,…, r. The sum of the ...exponential divisors of n is denoted by σ^((e)) (n). n is said to be an e-perfect number if σ^((e)) (n)=2n; (m;n) is said to be an e-amicable pair if σ^((e)) (m)= m+n =σ^((e)) (n); n_0, n_1, n_2,… is said to be an e-aliquot sequence if n_(i+1)=σ^((e)) (n_i)-n_i. Among the results established in this paper are: the density of the e-perfect numbers is .0087; each of the first 10,000,000 e-aliquot sequences is bounded.
We present a variety of numerical data related to the growth of terms in aliquot sequences, iterations of the function s(n) = σ(n) − n. First, we compute the geometric mean of the ratio s
k
(n)/s
k − ...1
(n) of kth iterates for n ⩽ 2
37
and k = 1, ..., 10. Second, we extend the computation of numbers not in the range of s(n) (called untouchable) by
Pollack and Pomerance 16
to the bound of 2
40
and use these data to compute the geometric mean of the ratio of consecutive terms limited to terms in the range of s(n). Third, we give an algorithm to compute k-untouchable numbers (k − 1st iterates of s(n) but not kth iterates) along with some numerical data. Finally, inspired by earlier work of
Devitt 76
, we estimate the growth rate of terms in aliquot sequences using a Markov chain model based on data extracted from thousands of sequences.
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