Diapositiva 1

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Aliquot Sequences
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Teoría de números
Muchos problemas abiertos de enunciado
sencillo (1) Conjetura de Golbach todo número
par es expresable como la suma de dos
primos. (2) Existen infinitos primos
gemelos. Fáciles de explorar mediante ordenador
Búsqueda de contraejemplos, estudio de
distribuciones, récords, ...
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Aliquot Sequences Sucesiones de sumas alícuotas o
sucesiones alicuatorias.
An aliquot sequence is a sequence of integers,
built with the sigma function ?(n) .
?(n) is the sum of divisors of an integer n
(include 1 and n).
Ejemplo ?(10) 1 2 5 10 17.
s(n) is the sum of the proper divisors is  s(n)
?(n) - n (s(n) es la suma de las partes
alícuotas de n). Ejemplo s(10) 1 2 5
7.
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Observemos que
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Aliquot Sequences Sucesiones alicuatorias
Iterate s(n), s(s(n)), s(s(s(n))) and so on.
Por ejemplo s(12) 16, s(16) 15, s(15)
9, s(9) 4, s(4) 3, s(3) 1, s(1) 1, s(1)
1, ... La sucesión alicuatoria que comienza
en 12 es por tanto 12, 16, 15, 9, 4, 3, 1, 1,
1, 1, 1, ...
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Perfect numbers s(n) n, s(6)
1 2 3 6. The smallest perfect numbers
are 6, 28, 496, 8.128, 33.550.336. Today there
are 43 perfect numbers known. (Se desconoce si
existen números perfectos impares). Abundantes
s(n) gt n, s(12) 12346 16.
Defectuosos s(n) lt n, s(22) 1211
14.
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Números amigos
Parejas de números (n, m) tq. s(n)
s(m) Pythagoras said true friendship is
comparable to the numbers 220 and 284 - this is
the smallest amicable pair s(220) 284 and
s(284) 220. Meanwhile thousands of amicable
pairs are found. Pedersen counted more than
10.410.218 (on December, 29th, 2005) (Se
desconoce si existen infinitos pares de amigos).
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Ciclos o números pandilla
Some cycles are of higher order (so called
sociable numbers).  Cycles with 4, 5, 6, 8, 9
and 28 members are known. Other orders are
possible, too. Today (May 2006) we know 146
aliquot cycles with higher order.  There are
138 cycles of the order 4, 3 of the order 6,
2 of the order 8, 1 of the order 5, 9 and 28.
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Números intocables de Erdös
n tales que n ? s(m) para todo m. Por ejemplo 2
y 5. Equivalente a "jardines del edén". En 1973
Erdös demostró que existen infinitos.
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Tipos de sucesiones
(1) La sucesión termina en 1 (siendo el número
anterior un primo). (2) La sucesión llega a un
número perfecto (y permanece constante). (3)
Llega a un par amigo o a un ciclo. (4) No está
acotada (?).
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Catalan Conjecture
This was first published by the Belgian
mathematician Eugène Catalan in the year
1887-88. Leonard Eugene Dickson extended in 1913
the so called Catalan conjecture "Each
aliquot sequence ends in one, in a perfect
number or in an aliquot cycle".Up to now it is
not possible to certify the Catalan conjecture.
Each confluence of two sequences gives some more
hope, but it's no proof of the conjecture, it's
only some work on the way to possible solution.
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To get a general idea a graphic presentation of
an aliquot sequence is helpful using a semi
logarithm axis, i.e. a linear x-axis beginning
with 0 for the index number of the sequence and
the y-axis on a scale of decadic logarithm for
the sum of the proper divisors. So there is a
function f N(n) -gt log10 s(n).There are three
types of aliquot sequences
(1) Terminating sequence
An ending aliquot sequence is a so called
terminating sequence. Normally the end of an
aliquot sequence is a prime (and 1).
The graph is a single irregular peak (ending in
a prime number)
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2) Cyclic ending
The graph ends in a horizontal line (ending in a
perfect number)
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Ending in an amicable pair (2620/2924)
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Ending in an aliquot 4-cycle
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Ending in the aliquot 28-cycle
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open-end sequence
Several sequences are not computed up to their
end. They are increasing and no one knows if
they will end or not. The smallest start-up
number (or key-number, beginning number) of such
a so-called open-end sequence is 276
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There are 5 open-end sequences in the interval
1, 1000 with the key numbers 276, 552, 564,
660 und 966.   There are 80 open-end-sequences
in the interval 1, 104. There are now 911
open-end sequences in 1, 105 and  9472
open-end-sequences in 1, 106. Any progress
in calculation can reduce these numbers.
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OE-sequence with deep minimum
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detailed table 1, 106  
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Normally an aliquot sequence ends in a prime.
Different sequences can come together and end in
the same prime. All these side sequences are
called a prime family. New calculations
occasionally lead to a confluence of two former
different aliquot sequences into one family.
About 1 of all integers are beginning numbers
(key numbers) of an open-end sequence. This is
an empirical result.
http//www.aliquot.de/aliquote.htm http//www.aliq
uot.de/index.htm
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Aliquot chain
Podemos reinterpretar las sucesiones
alicuatorias como cadenas dirigidas...
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Aliquot network
Estudio de las propiedades de la
red Distribución de conectividad Clustering Dist
ancia media dependiente del tamaño etc
Pero, añade algo nuevo ver la cuestión en
forma de red?
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E cjto. de escape
N 100
1
144
No importa
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E4
......
E3
......
E2
E1
2N
8N
N
4N
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E(N)
Forma funcional...
?
N
Apoyo a favor de la conjetura de Catalan
Apoyo en contra de la conjetura de Catalan
Evidentemente se puede intentar lo mismo con el
conjunto de números perfectos, amigos, ciclos e
intocables "relativos".
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The key element appears to be a problem termed
the ABC conjecture, which was formulated in the
mid-1980s by Joseph Oesterle of the University of
Paris VI and David W. Masser of the Mathematics
Institute of the University of Basel in
Switzerland.
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