Conjetura de Collatz (3n+1) variante

Question

Vamos a considerar el siguiente variante de Collatz (3n+1) :

si $n$ es impar, a continuación, $n \to 3n+1$

si $n$ es incluso entonces usted puede elegir : $n \to n/2$ o $n \to 3n+1$

Con esta definición, es posible construir un ciclo distinta de la trivial, es decir, $1\to 4 \to 2 \to 1$?

Saludos

Answer 1

Sí! Con el estándar de la conjetura de Collatz, cada número debe, finalmente, acabar en el ciclo de $4 \to 2 \to 1 \to 4 \cdots$ . Este ha sido verificada para todos los números hasta el $2^{60}$.

Con su definición alterada, usted puede comenzar a $2$, se aplican $3n+1$ en lugar de $n/2$, y luego continuar como el estándar de Collatz de nuevo.

$$2 \xrightarrow{3n+1} 7 \to 22 \to 11 \to \cdots ,$$ usted finalmente va a terminar en $2$ nuevo, ya que este es uno de los casos verificados.

Answer 2

$$7\to 22$$ $$22\to11$$ $$11\to34$$ $$34\to17$$ $$17\to52$$ $$52\to26\to13$$ $$13\to40$$ $$40\to20\to10\to5$$ $$5\to16$$ $$16\to8\to4\to2$$ $$2\to 3\cdot2+1=7$$

Answer 3

$4\to13\to40\to20\to10\to5\to16\to8\to4$

Answer 4

Hay muy pocos de esos ciclos. Aquí está la lista de todos los ciclos de longitud $\le 30$ a partir de cualquier número $< 10^5$ que nunca excederá $2^{63}$:

[2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4]
[2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 13, 40, 20, 10, 5, 16, 8, 4]
[2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 49, 148, 74, 37, 112, 56, 28, 85, 256, 128, 64, 32, 16, 8, 4]
[2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4]
[2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 61, 184, 92, 277, 832, 416, 208, 104, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4]
[2, 7, 22, 67, 202, 101, 304, 152, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4]
[4, 13, 40, 20, 10, 5, 16, 8]
[4, 13, 40, 20, 10, 5, 16, 8, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8]
[4, 13, 40, 20, 10, 5, 16, 49, 148, 74, 37, 112, 56, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8]
[4, 13, 40, 20, 10, 5, 16, 49, 148, 74, 37, 112, 56, 28, 85, 256, 128, 64, 32, 16, 8]
[4, 13, 40, 20, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8]
[4, 13, 40, 20, 61, 184, 92, 277, 832, 416, 208, 104, 52, 26, 13, 40, 20, 10, 5, 16, 8]
[5, 16, 8, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10]
[5, 16, 49, 148, 74, 37, 112, 56, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10]
[5, 16, 49, 148, 445, 1336, 668, 334, 167, 502, 1507, 4522, 2261, 6784, 3392, 1696, 848, 424, 212, 106, 53, 160, 80, 40, 20, 10]
[5, 16, 49, 148, 445, 1336, 668, 2005, 6016, 3008, 1504, 752, 376, 188, 565, 1696, 848, 424, 212, 106, 53, 160, 80, 40, 20, 10]
[7, 22, 11, 34, 103, 310, 155, 466, 233, 700, 2101, 6304, 3152, 1576, 788, 394, 197, 592, 296, 148, 74, 37, 112, 56, 28, 14]
[7, 22, 11, 34, 103, 310, 931, 2794, 1397, 4192, 2096, 1048, 524, 262, 131, 394, 197, 592, 296, 148, 74, 37, 112, 56, 28, 14]
[7, 22, 67, 202, 101, 304, 152, 76, 38, 19, 58, 29, 88, 265, 796, 2389, 7168, 3584, 1792, 896, 448, 224, 112, 56, 28, 14]
[7, 22, 67, 202, 101, 304, 152, 76, 38, 115, 346, 173, 520, 260, 130, 65, 196, 98, 49, 148, 74, 37, 112, 56, 28, 14]
[7, 22, 67, 202, 101, 304, 152, 76, 229, 688, 344, 172, 86, 43, 130, 65, 196, 98, 49, 148, 74, 37, 112, 56, 28, 14]
[8, 25, 76, 38, 19, 58, 29, 88, 44, 133, 400, 200, 100, 50, 151, 454, 227, 682, 341, 1024, 512, 256, 128, 64, 32, 16]
[8, 25, 76, 38, 19, 58, 29, 88, 44, 133, 400, 200, 100, 301, 904, 452, 226, 113, 340, 170, 85, 256, 128, 64, 32, 16]
[8, 25, 76, 38, 19, 58, 29, 88, 265, 796, 2389, 7168, 3584, 1792, 896, 448, 224, 112, 56, 28, 85, 256, 128, 64, 32, 16]
[8, 25, 76, 38, 115, 346, 173, 520, 260, 130, 65, 196, 98, 49, 148, 74, 37, 112, 56, 28, 85, 256, 128, 64, 32, 16]
[8, 25, 76, 229, 688, 344, 172, 86, 43, 130, 65, 196, 98, 49, 148, 74, 37, 112, 56, 28, 85, 256, 128, 64, 32, 16]
[10, 31, 94, 47, 142, 71, 214, 643, 1930, 965, 2896, 1448, 724, 362, 181, 544, 272, 136, 68, 34, 17, 52, 26, 13, 40, 20]
[11, 34, 17, 52, 26, 13, 40, 20, 61, 184, 92, 277, 832, 416, 208, 625, 1876, 938, 469, 1408, 704, 352, 176, 88, 44, 22]
[11, 34, 17, 52, 26, 13, 40, 121, 364, 182, 91, 274, 137, 412, 1237, 3712, 1856, 928, 464, 232, 116, 58, 29, 88, 44, 22]
[11, 34, 17, 52, 26, 13, 40, 121, 364, 182, 547, 1642, 821, 2464, 1232, 616, 308, 154, 77, 232, 116, 58, 29, 88, 44, 22]
[11, 34, 17, 52, 26, 13, 40, 121, 364, 1093, 3280, 1640, 820, 410, 205, 616, 308, 154, 77, 232, 116, 58, 29, 88, 44, 22]
[11, 34, 17, 52, 26, 79, 238, 119, 358, 179, 538, 269, 808, 404, 202, 101, 304, 152, 76, 38, 19, 58, 29, 88, 44, 22]
[11, 34, 17, 52, 157, 472, 236, 118, 59, 178, 89, 268, 134, 67, 202, 101, 304, 152, 76, 38, 19, 58, 29, 88, 44, 22]
[11, 34, 17, 52, 157, 472, 236, 118, 355, 1066, 533, 1600, 800, 400, 200, 100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22]
[11, 34, 17, 52, 157, 472, 236, 709, 2128, 1064, 532, 266, 133, 400, 200, 100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22]
[13, 40, 20, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 61, 184, 92, 277, 832, 416, 208, 104, 52, 26]
[13, 40, 20, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 241, 724, 362, 181, 544, 272, 136, 68, 34, 17, 52, 26]
[13, 40, 20, 61, 184, 92, 277, 832, 416, 208, 104, 52, 26]
[14, 43, 130, 65, 196, 98, 49, 148, 74, 37, 112, 56, 28]
[14, 43, 130, 65, 196, 98, 49, 148, 74, 37, 112, 56, 28, 85, 256, 128, 64, 32, 16, 49, 148, 74, 37, 112, 56, 28]
[16, 49, 148, 74, 37, 112, 56, 28, 85, 256, 128, 64, 32]
[19, 58, 29, 88, 44, 22, 67, 202, 101, 304, 152, 76, 38]
[19, 58, 29, 88, 44, 133, 400, 200, 100, 50, 25, 76, 38]
[20, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40]

Fue generado por el siguiente programa en Java:

public class ModifiedCollatz {

    int maxLength = 30;

    long[] currentPath = new long[maxLength];

    void find() {
        for (int i = 2; i < 100_000; i++) {
            find(i, i, 0);
        }
    }

    void find(long a, long goal, int depth) {
        if (depth >= maxLength || a < goal) {
            return;
        }

        if (depth > 0 && a == goal) {
            System.out.println(Arrays.toString(Arrays.copyOf(currentPath, depth)));
            return;
        }

        currentPath[depth] = a;
        if (a % 2 == 0) {
            find(a / 2, goal, depth + 1);
        }
        find(3 * a + 1, goal, depth + 1);
    }

    public static void main(String[] args) {
        new ModifiedCollatz().find();
    }
}

Answer 5

$$8 \rightarrow 25 \rightarrow 76 \rightarrow 38 \rightarrow 19 \rightarrow 58 \rightarrow 29 \rightarrow 88 \rightarrow 44 \rightarrow 22 \rightarrow 11 \rightarrow 34 \rightarrow 17 \rightarrow 52 \rightarrow 26 \rightarrow 13 \rightarrow 40 \rightarrow 20 \rightarrow 10 \rightarrow 5 \rightarrow 16 \rightarrow 8$$

Y otro:

$$16 \rightarrow 49 \rightarrow 148 \rightarrow 74 \rightarrow 37 \rightarrow 112 \rightarrow 56 \rightarrow 28 \rightarrow 14 \rightarrow 7 \rightarrow 22 \rightarrow 11 \rightarrow 34 \rightarrow 17 \rightarrow 52 \rightarrow 26 \rightarrow 13 \rightarrow 40 \rightarrow 20 \rightarrow 10 \rightarrow 5 \rightarrow 16$$

Tanto evitar la 4, 2, 1 ciclo.