4. Häufigkeit bei Collatz Folgen

Häufigkeit bei Collatz Folgen

  • E

    Ich habe mir die Häufigkeit H(n) bei der Collatz-Folge bei großen Zahlen angeschaut und festgestellt, dass z.B. zwischen 10^160 + 1 und 10^160 + 28620000 nur drei H(n), also Folgelängen, auftreten, zwischen denen gewechselt wird: 3407, 4069, 4617
    Bei 10^160 + 28620000 kommt dann 4922 und bei 10^160 + 41732001 eine Folge mit 4007 Elementen hinzu. Das finde ich wirklich überraschend, da doch teilweise von chaotischem Verhalten gesprochen wird. Hier zeigt sich doch eher eine stringente Ordnung.

    H(n) ist Anzahl von der Startzahl bis 1 (inclusive). Übrigens sind bei gleichen Schrittzahlen auch die Zahl der Multiplikationen (3x+1) und die Zahl der Divisionen (x/2) gleich!

    Beispiel:
    Zahl: 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 S: 4616 M: 1580 😨 3036

    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    Zahl: 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003 S: 4616 M: 1580 😨 3036

    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    Gibt es hierzu eine Studie? Zum Beispiel diese Darstellung für sehr hohe Zahlenbereiche: http://upload.wikimedia.org/wikipedia/en/thumb/9/9b/Totalstoppingtime.png/300px-Totalstoppingtime.png

    0
  • M

    Keine Ahnung 🙂
    aber ich vermute mal, dass das es schwieriger ist zu zeigen dass häufigkeit gegen 1 geht, als dass die Folge der H(n) beschränkt ist. Aber diese Frage ist trivial 🙂 nicht aber die antwort... (also der beweis)

    habe selbst wegen studium leider keine zeit mich intensiver mit der folge zu beschäftigen. Finde aber interessant das Problem.

    Kennst du The Dynamical System Generated by the 3n+1 Function | ISBN: 9783540639701? vielleicht steht da was drüber.

    0
  • E

    Ich habe folgende Überlegung angestellt:

    Zahlen wie 100000000001 kann man sich ja als zahl = a + b vorstellen.
    Interessant wäre doch nun, wie sich das Verhältnis Multiplikationen/Gesamtschritte verhält, wenn man Startzahlen wie folgt aufeinander folgen lässt:

    a      +b
10*a   +b
100*a  +b
1000*a +b

    also jeweils mit einem delta = 9*a zwischen den aufeinender folgenden Zahlen arbeitet.

    Hierbei wäre es interessant, wie sich Multiplikationen/Gesamtschritte in Abhängigkeit von log10(a) und b (als zweiter Parameter) verhält.

    Es fängt z.B. so an:
    Durch den 1-4-2-... Zyklus ergibt sich MDD als repetitives Element bis zum "Zusammenstoß" 😃 von a und b. Je größer die Startzahlen werden, jelänger kann sich diese repetitive Teilfolge "ausleben".

    MDDMDDMDDMDMDMDMDMDMDMDDDMDDDMDDMDMDDMDMDDDMDDMDDMDDDDMDMDMDDDDMDDMDDDDDDMDDMDMDDDMDDDDMDDDMDDDDDDMDMDMDDMDDDMDDDD
Zahl: 10000001
S: 114 M: 35 D: 79 M/S: 0.307018

    ...

    MDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDDMDMDDMDMDMDDDMDDDDDMDMDMDDDMDDMDDMDMDDDDDMDDMDMDDMDDMDMDDMDMDDDDMDDMDDMDMDMDMDMDDDMDMDDMDDMDDMDMDMDMDMDDMDDMDDMDDDMDDDMDDMDDDMDDMDMDMDDMDMDDDMDDMDDDDDDDMDMDMDDDMDDMDDMDMDMDDDMDDDMDDDDMDDMDDMDDDDDMDMDMDMDDMDDMDMDDMDDDMDMDMDMDMDMDDMDMDDDMDMDMDDMDDDMDDDDMDMDDMDDMDMDMDDMDMDDDMDMDDDDMDDMDDDMDMDDDMDDDDDDDMDMDMDDDDDDDMDMDDDDDDDMDMDMDDDMDDMDMDDMDMDMDDMDDMDDMDDDDMDDMDMDDDDMDMDMDDMDMDMDDMDDDDDMDDMDMDDMDMDDMDDDDDDMDDMDMDMDDDDMDMDDDDDDMDMDMDMDDDDDDMDDMDMDDDMDDMDMDMDDDDDMDDMDDMDDMDDMDDMDDMDMDDDDDDDMDMDDMDMDDDMDDDDMDDMDMDMDMDDDDMDMDMDMDDMDMDMDMDMDDDMDDMDDMDDDMDMDMDDMDDDMDDDDMDDMDDDDDMDDMDMDMDDMDMDDDMDMDMDDDMDDMDMDMDDMDMDDDDMDMDDMDMDMDMDMDMDMDDDDMDMDMDMDDDDMDMDMDDMDMDDDMDDDDMDDMDDMDDMDMDMDMDMDMDDDMDMDMDMDMDMDMDDMDDDDDDMDDDMDDMDDDDMDDDMDDMDMDDMDMDDMDDMDMDDMDMDMDMDMDMDMDMDDDMDMDDMDMDMDDMDMDDDMDDDMDDMDDDMDDMDMDDDMDDMDDMDDDMDDDDDMDMDMDDDMDMDDMDDMDDDDDMDMDDDMDMDDMDDDDMDDMDMDDMDDDDDDDDDMDDDMDDMDDDMDDMDMDMDDDDMDDMDDMDMDMDDMDDMDMDMDMDMDDMDMDMDDDMDDMDDDDDDMDDMDDDDMDMDDMDMDDMDDDDDDMDDDMDDMDDMDMDDMDDDDDMDMDMDDDMDDMDMDMDMDDMDMDMDMDMDMDMDDDDMDMDDDMDDDDDDDDDMDMDMDMDDMDMDDMDDDMDMDDDMDDDMDMDDDDDMDMDDMDMDDDMDMDMDDMDMDMDDDDDDMDMDDMDDMDDMDDMDDDDDMDMDDMDMDDDMDDMDDMDDMDMDMDMDDDDMDDMDMDDMDMDMDDMDDDMDDMDDDMDMDMDMDMDMDDMDDMDDMDMDDDMDMDMDDDMDDDMDDMDDMDDDDMDMDDDMDDMDDMDMDMDMDMDDMDMDDMDMDMDMDMDMDMDMDDMDMDMDMDMDDMDDDDMDMDMDDDDMDMDMDMDMDMDMDDDDMDMDMDMDDMDMDMDMDDDMDMDMDMDDMDDMDDMDDDDMDMDDMDDMDDMDMDDDDDMDDMDDDMDDMDMDMDDMDDMDDMDDDMDMDDMDMDMDDMDDMDMDDMDMDMDDMDMDMDMDMDDMDMDMDDMDDDMDDDMDDMDMDMDMDMDDDMDMDMDDDDDDMDMDDDMDDMDDMDDMDDDDDMDMDDDMDDDMDDDMDDDDDDMDDDDMDDDMDDDMDDDDMDDMDDMDDMDMDDDMDDDDDDDDDDMDMDDDDMDMDMDMDDDDMDDMDDDDMDDMDMDDMDDMDMDMDMDMDMDDDDDDMDMDDMDMDMDMDMDMDMDDDMDMDDMDDDMDDDMDDMDDMDDMDMDMDDMDMDMDMDMDMDMDMDDDDMDDMDMDDMDMDDDMDMDMDDMDDMDDDMDDMDMDDDDMDDDMDMDDDMDMDMDDDMDDMDDMDMDMDMDMDMDDMDDMDMDDDDDMDDDDMDDMDDMDMDMDDMDMDMDMDMDMDDMDDMDDDMDMDDDDDDMDDDDDDDMDDMDDMDDMDMDDMDDMDMDMDMDMDMDDMDMDMDMDDMDMDDDMDMDDDMDDMDMDMDMDDMDDMDMDMDMDMDMDMDMDMDDDDDMDMDMDDDMDDMDMDMDDMDMDDDMDMDDMDDMDDDDMDDMDMDDMDDDMDMDMDDDDDDMDMDDDDDDMDDMDMDDMDMDDMDMDDDMDDDMDMDDMDDMDDMDDMDMDDMDDDMDDMDDMDMDDDMDMDDMDDMDDDMDDMDMDDDDMDDMDDDDDDDDDMDDMDDDMDDDDMDDDMDDDMDMDMDMDDMDDMDDMDDMDDDMDMDDDMDDDMDMDMDMDDDMDDDDDDDDDMDMDDDDDMDDDDDMDDMDMDMDDMDDMDDDDDDMDMDMDMDDMDDDMDDDMDDDDMDDMDDDDDDDDDDMDDDDDDMDDDMDDDDMDDDMDMDMDDMDDDDMDMDDDDMDMDDMDDDDMDDDDDMDDDMDDDDDDMDMDMDMDDMDMDDMDMDDDMDMDMDDMDDMDDDMDMDDDMDMDMDDDDDMDDMDDMDDMDMDMDMDDMDDMDDMDMDMDDMDDMDMDDMDMDMDDMDMDDMDMDDDDDDDMDDMDMDMDMDDDDMDMDMDDMDMDDDMDMDMDDDMDDDMDDMDMDMDDDDMDMDMDMDMDMDDMDMDDMDDDMDDMDDDDDDMDDDDDDMDMDDDMDDDMDMDMDDDMDDDMDMDDMDMDMDMDMDMDMDMDMDMDMDDMDDMDMDMDDDMDMDDMDMDMDMDMDDMDMDMDMDMDMDMDMDMDDMDMDMDDMDMDMDDMDDDMDMDMDMDDDDMDDDDMDMDMDDDMDMDDMDDDDMDMDMDMDMDMDDMDMDMDMDMDMDDDMDMDMDDMDDMDMDDDMDDMDDDDMDDDDMDDDDDDMDDDMDDMDMDMDMDMDMDMDMDMDDMDMDMDMDDMDMDDMDMDMDMDMDDDMDDDDMDMDDDMDMDDMDMDMDMDMDDMDMDDDMDDMDMDMDMDDMDMDMDDDMDMDMDDMDDDMDDMDDMDMDMDMDMDDDMDDMDDDDMDMDMDMDDMDMDMDDDDMDMDDMDDMDDDDDDDDMDMDDDMDMDMDDMDDMDDDMDMDMDMDDDMDMDDMDMDMDMDDMDDDMDDDDDDDMDDMDMDDMDDDDDMDDDMDDMDDMDDDMDMDMDMDMDMDMDMDMDMDMDMDDDDDDMDMDDMDDMDDDDMDMDMDDMDMDDDMDDMDDMDDMDMDMDMDMDMDDDDMDDDDDDDDDDMDMDDMDDDDDDDDMDDMDDMDDMDDMDMDDDMDDMDDMDDDMDMDDDDMDDMDMDDMDDDMDMDDDMDDMDDDDDMDDDDDDDDMDDMDMDMDMDDMDMDMDDDDDDMDMDMDDMDMDMDMDMDDMDDMDMDDDMDMDMDMDDDMDMDMDDMDMDMDDMDDDDMDMDMDMDMDDDMDDDDDDMDDDDDMDMDMDMDDDMDDDMDDDMDDDDDDDDDDMDMDMDMDMDMDDDDMDMDMDMDMDMDMDMDMDDDMDDMDMDMDDDDMDDMDMDDMDDMDMDDMDMDDMDMDDDMDMDMDDMDDDMDDDMDMDMDMDMDMDMDMDMDDDDDMDDDMDMDMDDDDDDDDDDMDMDMDDMDMDDDDMDDDDDMDMDDMDMDDMDMDMDDMDDMDDDDDDDDDMDDMDMDMDMDMDDDMDMDMDMDMDMDDMDMDDDMDDDMDMDDDDDDDDMDMDMDDDDMDMDDDMDMDMDDDMDMDMDDDDDMDMDMDDMDMDDDDDDDMDDDMDMDMDDDMDDMDDDMDDMDMDDDDDMDMDMDDMDDDMDMDMDMDDMDMDMDMDMDMDDMDDDDDDDMDMDMDDMDMDMDDMDDMDDDMDDMDMDDDMDMDMDM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DDMDMDMDDDDDMDDDMDDMDMDMDDDDMDDMDMDDDDDMDDDDMDDMDDDDDDMDMDMDDMDMDDDMDMDMDMDMDMDMDMDMDMDDDDDDDMDDMDMDMDDDDMDDDDDMDMDMDDMDDDDDDMDDDDDMDMDDMDMDDDMDDDMDDMDDDDDDMDMDDDDDDMDMDMDDMDDMDDDDMDMDDDDMDMDMDMDDMDDMDMDDDMDDDMDMDMDDMDDDDMDDDDMDDMDDMDMDDMDDMDDDDMDMDMDMDDMDMDDDDDDDDDDMDDMDDDDDDDMDMDMDMDMDMDMDMDMDMDDDDMDDDMDMDMDDMDDMDDDMDDMDMDDDMDDMDMDMDMDDDMDDMDMDMDDMDMDMDMDMDDDDDMDMDMDMDDDMDDDDDDMDDDMDDDMDMDDMDMDDDDMDMDDDDMDMDMDDMDDDDDDMDMDMDMDMDDMDDDDDMDDDMDDMDDDMDMDDMDMDMDMDDDDMDDMDDMDMDMDMDMDMDMDDMDDDDDDDMDMDDDDDMDDMDDDMDMDDDMDMDDMDMDDDMDDMDMDDDMDMDMDDMDMDDMDDDMDDMDDDMDMDMDMDMDMDMDDDDMDMDDDMDDMDDMDMDDMDDDMDMDMDMDMDMDDMDMDMDDDDMDMDDMDMDMDMDDMDDDMDDMDMDMDMDDDMDMDMDMDMDMDMDMDDDMDDDDMDDDDDDMDMDDDDMDDMDMDMDMDDDMDMDDDDMDMDDDDMDDMDMDMDDDDDMDDDMDDMDMDDMDMDDMDDMDMDDDDDDDMDMDMDMDDMDMDDMDMDDDDDDDMDDDMDMDMDMDDDMDDDMDDMDDDMDDDMDDDDDMDMDDDMDDMDDMDDMDDDMDMDMDDDDDDMDMDDMDDDMDDDMDMDDMDMDMDDDMDDDDDMDDMDDMDMDDMDMDDDDMDDMDMDDMDDDDDDDDDDMDMDDMDMDDMDDMDMDMDDDMDMDMDMDDDDDDDMDDDDDDMDMDMDMDMDMDMDMDMDMDMDMDDDMDMDMDDMDMDMDDMDDMDDDMDMDMDMDDDDMDDDMDMDDMDMDDDDMDMDDMDMDMDMDDDDDMDDMDDDMDMDMDDMDDDDDDDDMDDMDDMDMDMDDMDDDDMDMDMDMDDMDMDDMDMDDDDMDDDDMDDMDDMDDMDDDDDDDMDDDDDDMDMDMDDMDDMDMDMDMDDMDDDDDDDDMDMDDMDDMDMDMDDDDDDMDDDDMDMDDMDDDDDDMDDDDMDDDDMDMDMDDMDMDDMDDDMDDDDDDDMDMDDMDDDDMDDMDDMDMDMDMDDMDMDMDMDMDMDMDDDMDMDMDMDMDDMDDMDMDDMDDMDMDMDDMDMDDDDMDMDMDMDDDMDMDMDDDDDMDDMDDMDMDDMDDMDMDDDDMDDMDDMDDMDDDMDMDDMDMDDDMDMDMDMDMDMDMDDDMDMDDMDDDDDMDDDMDMDMDDMDMDMDDMDMDMDDDMDMDDMDMDDDMDDMDDMDMDDMDDMDDDDMDDMDMDMDDMDDMDDMDDDMDDMDMDMDMDMDMDDDDDMDDDMDMDMDDMDDMDMDDMDDDMDDDMDDDMDMDDMDMDDMDMDMDDMDMDDDDMDDMDMDDMDMDMDMDMDDDMDMDMDMDDDMDMDDDMDMDDMDDDDDMDDDMDMDMDDDDMDMDDMDMDMDDDMDDMDMDDDMDDMDMDDDMDMDMDDDMDDMDMDDDDMDMDDDDMDDDMDMDDMDDDMDDDDDMDMDDMDDMDDDDDDMDMDMDMDDDDDDMDMDDDDMDMDMDDMDMDMDDDMDMDDDMDMDMDDDMDMDDDDMDMDDMDDMDDMDMDDMDDDMDMDMDDDDDMDMDMDDMDDDDMDDDDMDDDDDMDDDMDDDDMDDDDDMDMDMDDMDMDMDMDMDMDMDMDMDDMDMDDDMDMDDMDMDDDMDMDMDDDDDDDMDMDMDDDMDDMDDMDDDDDDDDMDMDMDDDDMDDDDDMDMDDMDDDDDDMDDDMDMDDMDDMDDMDDMDDMDDDDDMDDMDMDMDDMDDDMDMDMDDDMDDMDDDMDMDMDMDDDDDMDDDMDMDDDDMDDDMDMDMDMDDDDMDMDMDMDMDMDDDMDMDMDDMDDMDDDMDDDDDMDDMDMDMDMDMDDDDDDDDMDDDMDDDDDMDMDMDMDMDMDDDMDMDMDMDDMDMDMDMDDDDDDDMDMDDDDDDDMDMDDDMDMDDMDMDDMDDDDMDMDMDDDMDDDMDMDDMDDMDDDMDDMDMDMDMDDDMDMDDDDDMDMDMDDDDDMDMDMDMDMDMDMDDDDMDMDMDMDDMDMDDDMDDMDMDMDDDDDDDMDMDDMDMDMDMDDDDMDMDDMDMDMDMDDDMDDMDDDMDMDMDDDMDDDMDMDDMDMDDDDMDDMDDMDMDDMDMDMDDDMDMDMDMDDDDDMDDMDMDMDDDDMDDMDDDMDMDDMDDMDDMDMDDDDMDDDDMDDDDDMDDDDMDMDDMDMDMDDDMDMDMDMDMDDMDDDDDMDDMDDDMDDMDMDMDDMDMDDDDDMDMDDMDDDMDMDMDMDMDMDDMDDMDMDDMDMDDMDMDMDMDDDMDMDDDDMDDDDMDDMDDMDMDMDDMDMDDMDMDMDDDMDDDMDMDDDDMDMDMDMDDMDDDMDDDMDMDMDDMDMDMDDDMDDDMDDMDMDDMDDMDDDDDMDMDDDDDMDMDMDDMDMDMDMDDMDDDMDDDMDDMDMDMDMDMDMDDDDMDMDMDDDMDDDMDDMDMDDDDDDDDDMDMDDMDDMDDDMDDDDDMDMDMDDMDDMDMDMDMDDDDDMDMDMDMDMDMDMDMDDDDDDMDDMDDDDDMDDMDDDDMDMDDMDDMDMDDMDMDDMDDMDDMDMDDDDMDDMDDDDDDMDMDMDDMDDMDDMDDDMDDDDMDDDMDDMDDDMDDDDDDMDDMDDDMDMDMDMDMDMDDDMDDDMDDMDDDMDMDDMDMDDMDDMDDDDMDMDMDMDMDDDDMDDMDMDDDDMDMDMDDMDMDMDDDMDDMDDDDDDDMDMDMDDMDDDDMDMDMDMDMDMDDDMDMDMDDDDMDMDDMDMDMDDDDMDDDDDMDMDDMDDDMDMDDDDDMDMDMDDDDMDDMDMDDMDMDDMDMDDDDDDDMDDMDDDMDMDDMDMDMDMDMDDMDMDDMDDMDDDMDMDDMDDMDDMDDDDDDMDDDDMDDDMDDDMDDDMDDDMDMDDMDDMDMDMDDMDMDMDDDDDDDDDMDMDMDMDDDMDDMDMDMDDMDDMDMDMDMDDMDMDDMDDDDMDDMDDDDDMDDMDMDMDMDMDMDMDDMDDMDMDDDDDDMDDMDMDMDDMDMDDDMDMDDDMDMDMDMDMDMDDDMDDDDMDDDMDDDDDMDDDDMDMDDMDDMDMDMDMDMDMDDMDDMDMDMDMDDDDMDMDMDDMDMDMDMDDDMDMDDDMDMDDDDMDDDDDMDMDDDMDMDMDMDMDDMDMDMDMDDDMDDDMDMDMDDDMDMDDDMDMDMDMDDMDDMDMDMDDDMDDMDDDDDMDMDMDMDDDMDDDDDMDDMDMDMDDMDMDMDDDDDDMDDMDMDMDDMDDDDMDDMDMDMDDDMDMDDMDMDDDDDDMDMDMDMDMDMDDDMDDDMDDDMDMDMDDMDDDDMDMDMDMDDDDMDDDMDDDDMDMDDDMDMDDMDDDMDDMDDDMDDMDDMDMDMDMDMDMDMDDDMDMDDDDDMDMDDDDDMDDMDMDDDDMDDDDMDMDMDMDDDMDMDMDDDMDDMDMDMDDMDMDMDDMDDDMDDMDMDMDDMDMDMDMDDDMDMDMDDMDDDDDDDDMDMDDMDDDDDDMDDMDDMDMDMDDMDDMDMDDMDDMDMDDMDDDMDMDDDMDMDDDDMDMDDMDMDMDMDMDDDMDDDMDDMDDMDDDDMDDMDDMDMDDDDMDMDMDMDDMDMDMDDDMDMDDMDMDMDMDMDDDMDDDMDMDDMDDMDMDMDDMDMDMDDMDDMDDDDDMDDDDDDMDMDDMDDMDDMDDMDDMDDDMDMDMDMDDDDMDDDMDMDDDMDMDDDMDMDDDDMDDDDMDMDMDMDMDDDDDDDMDDDMDMDMDMDDDDDMDDDDMDMDMDDMDMDMDDDMDDMDDMDDDDMDDMDMDMDDMDMDDMDDDMDMDDDMDDMDDMDDDMDDDDMDDDMDDDDDDMDDMDDDMDMDMDMDMDMDMDDMDMDMDDMDDMDDMDMDDDMDMDMDMDDMDMDDDMDMDDDMDDMDMDDMDMDMDMDMDMDMDDMDMDDDMDMDDDDDMDMDDMDDDDDMDMDDMDMDDMDMDMDDDDDDMDMDMDMDDDDMDMDDDDMDDDMDDDDMDMDDDDMDDDMDMDMDMDDMDMDMDMDDDMDMDMDDDDMDMDMDDDMDDMDMDMDDMDDMDMDMDDDDDDDMDMDDDMDMDMDMDMDMDMDMDDMDDDMDDMDMDDMDMDMDMDMDDMDDDMDMDDMDDMDMDDDMDDDDDDDMDMDDDMDMDDDMDMDMDDMDMDDMDDDDMDDMDDMDDDMDDDDDDMDDDMDDMDDDMDDMDMDDDDMDMDDDMDDDMDMDMDDMDDMDDMDDDMDMDMDMDMDMDMDMDMDDDDDMDMDDDDMDDDDDMDMDDDMDDMDMDDMDMDMDMDMDMDDDDDDMDDDDDMDMDDMDMDDMDMDMDDDMDDMDMDDMDMDDDDMDDDMDMDMDMDMDMDDDDDDDMDMDDDDMDMDDDMDMDDDMDDMDMDDMDDDMDMDMDDMDMDDMDDDMDDDDMDDMDMDDDDDMDDDDMDDMDDMDMDMDMDMDMDDMDDMDMDDMDMDMDDMDDDDMDDDDDMDMDDMDDDDMDDMDDDDMDMDMDMDMDDDMDDDDDDMDDMDDDMDDMDMDDMDDMDMDDMDMDMDMDMDMDMDMDMDMDMDDMDMDMDDDMDDDDDMDMDDMDDMDDDDMDDDDDMDDDMDMDDDMDMDMDDDMDMDMDDMDDMDDMDMDMDMDMDDDDMDDMDDDDDDMDMDDMDMDMDDMDMDDMDMDMDMDDDMDDMDMDDMDMDMDMDMDDMDMDMDDDMDDMDMDDDMDDMDDMDDMDMDDDMDMDMDMDMDDDMDDDDDDMDDMDDMDMDDMDMDMDDMDDDMDMDD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MDDMDMDMDMDDMDDDMDDMDDDMDDDDDDMDMDDDDDMDMDDDDDMDMDDDDDMDMDDDMDMDDDDMDDMDMDDDDDMDDDDDMDDMDDMDDDMDMDMDDDMDMDDDMDMDMDMDDMDDMDDMDMDMDMDDDMDDMDDDMDMDDMDDDMDDDMDDMDDMDDMDMDMDMDDMDDMDMDMDDDDMDDDMDDMDMDMDMDMDMDDDMDDDDDMDDDMDMDDMDDMDMDDDDMDMDMDMDMDMDMDDMDDMDDDDMDMDMDMDDMDDMDDMDDMDDMDDDDDDDDMDMDDMDDDDDDDMDMDMDMDDDDDMDDMDDMDMDDDMDMDMDMDDMDDMDDMDMDMDDDDDDDDMDDDDDDMDMDDMDDDDMDMDMDMDMDDMDDMDMDDMDDDMDMDDDDDMDDDMDMDDDDDMDMDMDDDMDDMDDDMDDDDMDDDDMDDDDDMDMDDDDMDDMDMDMDDDMDDDDMDMDMDDDMDMDMDDMDMDMDMDDMDDMDDDDDDDMDDMDMDMDDMDMDDMDMDDDDMDDDDDDMDDDMDDMDDDMDMDDMDMDDDMDDDMDDMDDDMDDDMDMDDDDDMDDDDDMDMDDDMDMDDMDMDDDMDDMDMDMDDMDMDMDMDMDMDMDMDDDDMDMDMDMDMDMDMDDDMDMDDMDDMDMDDMDDDDMDDDDDMDDDMDDDDDDMDMDMDMDDDMDDMDMDDMDMDDMDDMDDDDMDDDDMDMDMDMDDDDMDDMDDMDDMDMDMDDMDMDMDMDMDDMDDMDDDDDDMDDDMDMDMDMDMDMDDMDMDDDDDMDMDDMDMDMDDMDMDDMDDMDDDMDMDMDDDMDMDDMDMDDMDMDDDMDDMDDMDMDDDMDMDDMDDMDDDDDMDDMDMDMDDDDMDMDDDMDDDMDDMDMDMDDDDDMDDMDDMDDDDMDMDDDMDMDDDDMDDMDMDMDDMDDMDMDMDDMDDMDMDMDMDMDMDDDMDDMDDDDDDDMDDMDMDDMDMDMDMDMDMDMDMDMDMDMDMDDDDDDMDDDMDMDDDDMDDDMDMDDDMDMDMDDMDMDDMDMDDDMDDMDMDDDDDDMDMDDMDMDDDMDDMDMDDDMDDDMDDDDMDMDMDMDDMDMDMDMDMDDMDMDDMDDMDDMDDMDMDDMDMDMDDDMDMDDDDMDDMDMDMDDDMDDDMDMDMDMDMDMDMDDMDDMDDDMDMDMDMDMDMDDDMDDMDDDDDMDDDDMDDDMDMDMDMDMDMDMDDMDDMDMDDMDDMDMDMDDMDDMDMDDMDMDMDDMDDDMDDMDDMDMDMDMDMDMDMDDDMDDDDDMDMDMDMDMDMDMDDDDMDDMDDMDMDDMDMDMDDMDDDMDMDMDMDDDMDMDMDDMDDMDMDDMDDDMDMDDDMDMDMDDDMDDMDMDDMDMDMDMDMDMDDDDMDMDMDDDMDDDMDDMDDMDDMDMDMDDMDDMDDDDDDMDDDDMDDMDMDMDDMDMDMDDMDDMDDMDMDDMDMDMDMDDDDMDDDDMDDMDDMDMDDDMDMDDMDDMDDMDDMDDDMDDDDMDDMDDDDMDDDMDMDMDDDDDMDDDD
Zahl: 10 hoch 893 + 1
S: 21537 M: 7184 D: 14353 M/S: 0.333566

    Interessant finde ich, ob M/S (Multiplikation/Gesamtschritte) konvergiert und wie sich die MD-Ordnung (Multiplikation Division) verhält.

    Schaut man sich die Zahlenfolgen konkret an, so stellt man fest, dass a und b irgendwann "zusammenstoßen" (die 5 kommt von links rein)! Vorher kann b getrennt seine eigene Collatz-Folge am rechten Rand der Zahl spielen. Ab dem Zusammenstoß ändert sich die MD-Ordnung.

    Zahl: 10 hoch 922 +1
S: 22688 M: 7592 D: 15096 M/S: 0.334626

Zahl: 10 hoch 923 +1
S: 22239 M: 7417 D: 14822 M/S: 0.333513

Zahl: 10 hoch 924 +1
S: 23005 M: 7712 D: 15293 M/S: 0.335231

    Für b=1 ist M/S ca. 0.333, konvergiert im betrachteten bereich bis ca. 10 hoch 1000 nicht, stellt logischerweise aber "MDD", also 1/3 Multiplikation, dar.

    Manchmal findet man außergewöhnliche Zhalen wie:
    10^1016 + 1
    Delay: 24396 Even: 16264 Odd: 8132 O/D: 0.333333 Completeness(O/E): 0.5 Glide: 3 Strength: -8132 Residue: 1.004167355307
    Da fragt man sich, wieso genau diese Zahl diese Eigenschaften hat.

    0
Anmelden zum Antworten