Ups and Downs

My last post titled "The Path to Unity" focused on numbers that lead to zero when the recurrence shown in Figure 1 is applied:

Figure 1

Today, the number associated with my diurnal age (28090) leads to a sequence with wild gyrations but that eventually loops. Firstly though, the factorisation:$$28090=2 \times 5 \times 53^2$$The sequence has 164 members:

28090, 337080, 16179840, 112360, 2696640, 226517760, 73391754240, 67813980917760, 253895544556093440, 19590705598464, 6976747008, 3767443384320, 1453488960, 732558435840, 381540852, 82412824032, 95385213, 6867735336, 3296512961280, 1271802840, 488372290560, 308315840, 51797061120, 31078236672000, 72723073812480000, 6464273227776, 8144984266997760, 1414059768576, 1374466095055872, 4354308589137002496, 50161634946858268753920, 1209530163649167360, 69995958544512, 201588360608194560, 14284889498880, 3306687384, 1269767955456, 801621184, 134672358912, 80803415347200, 283619987868672000, 18757935705600, 60775711686144000, 4637951136000, 1431466400, 463795113600, 214719960, 61839348480, 80143795630080, 21405928320, 24775380, 4013611560, 7963515, 477810900, 1474725, 53090100, 11467461600, 15168600, 2184278400, 5056200, 546069600, 1685400, 121348800, 30579897600, 31460800, 3964060800, 2568711398400, 5340350997273600, 674286742080, 501701445, 63214382070, 27308613054240, 47189283357726720, 10923445221696, 10836751212, 2925922827240, 2738663766296640, 724514223888, 521650241199360, 172503386640, 113852235182400, 344289159191577600, 28980568955520, 66771230873518080, 12420243838080, 9952118460, 2149657587360, 3980847384, 573242023296, 192609319827456, 208018065413652480, 64203106609152, 63693558144, 221158188, 19904236920, 5732420232960, 4333709696117760, 9724844558088253440, 109229454076447262638080, 3010070934646364160, 307401034992480, 142315293978, 40986804665664, 48200482286820864, 260282604348832665600, 10299248351884800, 72300723430231296000, 2615043526845750, 9790722964510488000, 431689725066600, 2051389573516483200, 113061594660300, 46527405210, 25124798813400, 65123478524332800, 7444384833600, 21104831003256000, 1550913507000, 2679978540096000, 20260637763125760000, 577226147097600, 69411513600, 89957321625600, 24027062400, 27809100, 4505074200, 2270557396800, 1001127600, 2224728, 30899, 185394, 4449456, 400451040, 1112364, 60067656, 8649742464, 22525371, 946065582, 181644591744, 210236796, 45411147936, 70078932, 12614207760, 15016914, 1081217808, 259492273920, 266967360, 89701032960, 67955328, 353934, 12741624, 1528994880, 3033720, 291237120, 1078656, 22472, 269664, 9707904, 101124, 2730348, 50562, 2809, 8427, 50562

If we plot the sequence members, we get what is shown in Figure 2.

Figure 2

As can be seen, there are two massive spikes. The factorisations of both numbers are as follows:

• \(50161634946858268753920 = 2^{23} \times 3^{11} \times 5 \times 7^5 \times 11 \times 13 \times 53^2\)
• \(109229454076447262638080 = 2^{17} \times 3^{20} \times 5 \times 7 \times 11 \times 13 \times 17 \times 53^2\)

The associated number of divisors are:

• \(50161634946858268753920\) has \(41472\) divisors
• \(109229454076447262638080\) has \(36288\) divisors

These numbers of divisors of course divide into each of their respective numbers:

• \( \dfrac{50161634946858268753920}{41472}=1209530163649167360\)

• \( \dfrac{109229454076447262638080}{36288}=3010070934646364160\)

Figure 3 shows the trajectory plotted on a vertical logarithmic scales where the ups and downs are clearly visible and the difference in height between the the two spikes seems minor:

Figure 3: vertical logarithmic scale

The takeaway to all this is that, even though many sequences "blow up" under the rules imposed, some still manage to enter a loop despite having impressive spikes. Notice how the loop (50562, 2809, 8427, 50562) occurs:

• \(50562 = 2 \times 3^2 \times 53^2 \text{ with }18 \text{ divisors} \)
• \(2809 = 53^2 \text{ with } 3 \text{ divisors} \)
• \(8427 = 3 \times 53^2 \text{ with } 6 \text{ divisors} \)
• \(50562 = 2 \times 3^2 \times 53^2 \text{ with } 18 \text{ divisors}\)

\(2809\) is the minimum value reached and this is simply our original number (28090) divided by 10. Thus we have:$$ \frac{28090}{10}=2809 \text{ as the minimum value}$$Once again, the behaviour of the number 28090 (or any other number) under the above rules is INDEPENDENT of the number system being used.