La suite de Syracuse⚓︎

Définition

La suite de Syracuse est définie de la manière suivante :

u_{n + 1} = {\begin{cases} u_{n} / 2 & si u_{n} est pair \\ 3 u_{n} + 1 & sinon \end{cases}

Comme nous l'avons vu l'année dernière, il est conjecturé qu'il existe toujours un rang $n$ pour lequel $u_{n} = 1$ , pour n'importe quel entier strictement positif $u_{0}$ .

Plus d'informations ici :

Exercice 18: prochain(u)

Compléter le code de la fonction prochain qui prend en paramètre un entier non nul u et qui renvoie la valeur qui vient après u dans la suite de Syracuse.

>>> prochain(14)
7
>>> prochain(7)
22

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7
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9
10
def prochain(u):
    if ...:
        return ...
    else:
        return ...
# Tests
assert prochain(14) == 7
assert prochain(7) == 22
 
 
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>>>

x

Évaluations restantes : 5/5

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Exercice 19: syracuse(u)

Compléter le code de la fonction récursive syracuse qui prend en paramètre un entier non nul u et qui affiche tous les termes de la suite de Syracuse en partant de u.

>>> syracuse(5)
5
16
8
4
2
1

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def syracuse(u):
    print(u)
    if u > 1:
        syracuse(...)
 
 
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>>>

x

Évaluations restantes : 5/5

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Durée de vol

On appelle durée de vol de la suite de Syracuse la valeur du plus petit entier $k$ tel que $u_{k} = 1$ .

Par exemple, la durée de vol si $u_{0} = 5$ est de 5.

Exercice 20: duree_vol(u)

Compléter le code de la fonction récursive duree_vol qui prend en paramètre un entier non nul u et qui renvoie la durée de vol de la suite en partant de u.

>>> duree_vol(1)
0
>>> duree_vol(7)
16

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def duree_vol(u):
    ...
# Tests
assert duree_vol(1) == 0
assert duree_vol(7) == 16
 
 
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>>>

x

Évaluations restantes : 5/5

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L'altitude maximale

L'altitude maximale de la suite de Syracuse est la valeur maximale de la suite.

Par exemple l'altitude maximale en partant de 5 est 16.

Exercice 21: altitude_max(u)

Compléter le code de la fonction récursive altitude_max qui prend en paramètre un entier non nul u et qui renvoie l'altitude maximale de la suite en partant de u.

>>> altitude_max(11)
52
>>> altitude_max(32)
32

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def altitude_max(u):
    ...
# Tests
assert altitude_max(11) == 52
assert altitude_max(32) == 32
 
 
הההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההה
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>>>

x

Évaluations restantes : 5/5

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