such that is called the total stopping time of . If one of the indexes or doesn't exist, we say that the stopping time or the total stopping time, respectively, is infinite. The Collatz conjecture asserts that the total stopping time of every is finite. It is also equivalent to saying that every has a finite stopping time. Since is even whenever is odd, one may instead use the "shortcut" form of the Collatz function: This definition yields smaller values for the stopping time and total stopping time without changing the overall dynamics of the process. Empirical data For instance, starting with and applying the function without "shortcut", one gets the sequence 12, 6, 3, 10, 5, 16, 8, 4, 2, 1. The number takes longer to reach 1: 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. The sequence for , listed and graphed below, takes 111 steps (41 steps through odd numbers, in bold), climbing as high as 9232 before descending to 1. 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1 Numbers with a total stopping time longer than that of any smaller starting value form a sequence beginning with: 1, 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, ... . The starting values whose maximum trajectory point is greater than that of any smaller starting value are as follows: 1, 2, 3, 7, 15, 27, 255, 447, 639, 703, 1819, 4255, 4591, 9663, 20895, 26623, 31911, 60975, 77671, 113383, 138367, 159487, 270271, 665215, 704511, ... Number of steps for to reach 1 are 0, 1, 7, 2, 5, 8, 16, 3, 19, 6, 14,