Editing Collatz conjecture (section)

==Empirical data==
For instance, starting with {{math|''n'' {{=}} 12}} and applying the function {{math|''f''}} without "shortcut", one gets the sequence {{CSG|12}}.

The number {{math|''n'' {{=}} 19}} takes longer to reach 1: {{CSG|19}}.

<!-- NOTICE TO EDITORS...  Please note that the number of steps is one less than the number of elements of the sequence!  So the number of 111 steps is CORRECT for n=27.  Thanks for paying attention to this factoid! -->
The sequence for {{math|''n'' {{=}} 27}}, listed and graphed below, takes 111 steps (41 steps through odd numbers, in bold), climbing as high as 9232 before descending to 1.

: {{CSG|27|bold= odd}}{{OEIS|id=A008884}}

[[File:Collatz5.svg|frameless|upright=2|center]]

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, ... {{OEIS|A006877}}.

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, ... {{OEIS|id=A006884}}

Number of steps for {{mvar|n}} to reach 1 are
:0, 1, 7, 2, 5, 8, 16, 3, 19, 6, 14, 9, 9, 17, 17, 4, 12, 20, 20, 7, 7, 15, 15, 10, 23, 10, 111, 18, 18, 18, 106, 5, 26, 13, 13, 21, 21, 21, 34, 8, 109, 8, 29, 16, 16, 16, 104, 11, 24, 24, ... {{OEIS|id=A006577}}

The starting value having the largest total stopping time while being
:less than 10 is 9, which has 19 steps,
:less than 100 is 97, which has 118 steps,
:less than 1000 is 871, which has 178 steps,
:less than 10<sup>4</sup> is 6171, which has 261 steps,
:less than 10<sup>5</sup> is {{val|77031}}, which has 350 steps,
:less than 10<sup>6</sup> is {{val|837799}}, which has 524 steps,
:less than 10<sup>7</sup> is {{val|8400511}}, which has 685 steps,
:less than 10<sup>8</sup> is {{val|63728127}}, which has 949 steps,
:less than 10<sup>9</sup> is {{val|670617279}}, which has 986 steps,
:less than 10<sup>10</sup> is {{val|9780657630}}, which has 1132 steps,<ref>{{cite journal |last1=Leavens |first1=Gary T. |last2=Vermeulen |first2=Mike |date=December 1992 |title=3''x'' + 1 search programs |journal=Computers & Mathematics with Applications |volume=24 |issue=11 |pages=79–99 |doi=10.1016/0898-1221(92)90034-F|doi-access= }}</ref>
:less than 10<sup>11</sup> is {{val|75128138247}}, which has 1228 steps,
:less than 10<sup>12</sup> is {{val|989345275647}}, which has 1348 steps.<ref name=Roosendaal>{{cite web |last=Roosendaal |first=Eric |title=3x+1 delay records |url=http://www.ericr.nl/wondrous/delrecs.html |access-date=14 March 2020}} (Note: "Delay records" are total stopping time records.)</ref> {{OEIS|id=A284668}}

These numbers are the lowest ones with the indicated step count, but not necessarily the only ones below the given limit. As an example, {{val|9780657631}} has 1132 steps, as does {{val|9780657630}}.

The starting values having the smallest total stopping time with respect to their number of digits (in base 2) are the [[Power of two|powers of two]], since {{math|2<sup>''n''</sup>}} is halved {{mvar|n}} times to reach 1, and it is never increased.