3x+1 Strength and Strength Records

The Strength S of a number N is defined as S(N) = 5.O - 3.E, where O and E represent the number of Odd (3x+1) and Even (x/2) steps respectively.

Note that the Strength can be defined equally well when the algorithm is defined using (3x+1)/2 rather than 3x+1. In that case again let O' and E' be the number of Odd and Even iterations, and we have S(N) = 2.O' - 3.E'. This makes the Strength independent of the choice of algorithm.

It should be obvious that any Strength record is necessarily a Class Record as well. But the Strength turns out to be fairly constant over large ranges of Class Records. So much so that only a few Strength records are known. Current data suggest they are roughly logarithmically distributed and occur only once in every ten powers of two or thereabout.

Depending on the exact definition of the 3x+1 function the first Strength record is either 1 or 2. The number N=1 has a Delay of zero, and therefore also Strength zero. If one does not want to see 1 as the first record the first record is 2, with a S(2) = -3. Since either number is rather trivial the first Strength record in the table below has been labeled as #0. It is rather surprising to see that neither value is surpassed by any N < 60,000,000.

As the table shows the first non-trivial Strength record has 8 digits already, thus emphasizing their rarity. For many years only four Strength records were known with certainty. In 2011 the results of the distributed search eventually confirmed Strength record number five.
The other numbers depicted are current best known candidates for the next records. At this moment these are the only candidates known for N < 2100. The smallest of these, the candidate for record number six, may just be reachable with the current distributed project, but the next candidates are so large we may never be able to confirm them unless a totally new approach can be found.

(Potential) Strength records currently known < 2100
# Strength Delay Number
12? 69 3825 31549,861135,742690,018597,749695
11? 65 3661 2421,645885,513162,728286,680347
10? 60 3316 4,783763,851719,920612,626591
9? 58 3154 239958,815015,359848,833791
8? 56 2968 7219,136416,377236,271195
7? 55 2955 6852,539645,233079,741799
6? 46 2710 268,360655,214719,480367
5 38 2254 104899,295810,901231
4 28 1820 100,759293,214567
3 17 1549 3,743559,068799
2 10 1210 13371,194527
1 9 949 63,728127
0 0 0 1
Although numbers with high Strength appear to be very rare it may be that the Strength values are unbound. Thus the following Conjecture may be true:
Strength Conjecture:
For every integer S a positive integer N exists with Strength(N) = S
For S ≤ -5 this is fairly easy to prove. Consider for instance that a number of the form 2n has Strength -3n. Also S(2N) = S(N) - 3, and since S(3) = -5 and S(5) = -7 this covers all numbers below -5.

Therefore numbers with Strength -5; -6; -7 and so on are simply 3; 4; 5; 6; 8; 10; 12; 16; 20; 24; 32; 40 and so on.

For S ≥ -4 however the situation is far less obvious. For instance, the smallest numbers with Strengths 0; -1; -2; -3 and -4 are 1; 209,303017; 235,465895; 2; and 179,008815 .

There may exist numbers with very high Strengths though, since the list of Class Records shows slowly increasing Strength values. Indeed, it appears that for every Strength S there is a limit beyond which all Class Records have higher Strengths. The evidence is rather thin, but the table below depicts the highest Class Records with Strength S.

Highest Known Class Records with particular Strength (>= -8)
Strength Delay Number
-8 920 72945,377791
-7 1237 1,024337,865708
-6 1274 1,726869,407806
-5 1375 11,787870,157403
-4 1484 95,427878,556519
-3 1473 60,387954,399047
-2 1486 63,618585,704351
-1 1523 112,265328,502742
0 1528 100,197749,947111
1 1525 74,843552,335161
2 1698 2421,269892,238985
3 1719 2933,495759,934703
4 1724 2687,265553,316295
5 1721 2034,106312,229086
6 1742 2510,719179,521742
7 1739 1883,039384,641307
8 1776 3275,533932,696303
9 1925 60614,485209,289967
10 1882 20089,765945,518247
11 1951 69520,368888,759751
12 2076 766624,412109,824743
13 2113 1,345161,942161,756487
14 2142 2,029435,401161,373768
15 2131 1,284252,089797,431844
16 2192 3,749740,882243,642695
17 2165 1,694294,236575,537470
18 2234? 5,780540,886094,271145
As can be seen the Strength for Class Records is increasing slowly, and by a rough approximation the highest Class Record with a particular Strength is somewhere around 1500 + 40S. There are indications that this trend is persisting with higher Class Records as well.

Thus we may venture

Increasing Class Record Strengths Conjecture:
Let CR be a Class Record for Delay D. Then for every Strength s a limit DMAX(s) exists with
D > DMAX(s) => S(CR(D)) > s
Example :
CR(1925) has a Strength of 9. All Class Records currently known beyond 1925 have higher Strengths.
Therefore DMAX(9) = 1925, and the conjecture states that if D > 1925 then the Strength of CR(D) will be > 9.

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