Problems & Puzzles: Puzzles

Puzzle 67.- The P/M-SOPF sequences

I have introduced a slight change of the SOPF sequences proposed by Patrick De Geest (see Puzzle 66):

To add the Sum of the Prime Factors if C_i is even, or to subtract the Sum of the Prime Factors if C_i is odd_i is even, or to subtract the Sum of the Prime Factors if C_i is odd. I will call these types of sequences "The Plus/Minus SOPF sequencesMinus SOPF sequences", or simply The P/M-SOPF sequences

Just in order to be precise, let C_i =p₁^a1.p₂^a2… , then:

a)     C_i+1 = C_i + (sum of) (a_j . p_j), if C_i is even

b)     C_i+1 = C_i - (sum of) (a_j . p_j), if C_i is odd

As usual these sequences will end whenever a prime number is reached. Example: 64, 76, 99, 82, 125, 110, 128, 142, 215, 167

The obvious question is, will ever happen closed loops?. The answer is affirmative. An example of closed loop is : 650, 675, 656, 705, 650, …

675 = 650 + (2+5+5+13); 656 = 675 - (3+3+3+5+5);

705 = 656 + (2+2+2+2+41); 650 = 705 - (3+5+47)

In the following table you can see all the closed loops that I have obtained below 99999.

Would you like to find the least closed loops for lengths 5, 6, 7 and larger than 8?

Jack Brennen has solved completely this puzzle. Here are his results (4/10/99):

"Loops of length 5:

388770 401739 391422 392249 392001
594320 594392 594978 596907 594585
4530734 4705008 4705123 4604967 4601545

Loops of length 6:

678648 706934 708468 709891 709573 678699
1647682 2471525 2470992 2522482 2523039 1682023
3153838 3173880 3200343 3198258 3206737 3203135

Loop of length 7:

246670 248145 247833 247620 251759 250559 246959

Loops of length 8:

674098 1011149 967163 956207 819599 745079 741983 674519
908228 913110 915898 928314 945516 945595 918566 919593
2204614 3306923 3303167 3296050 3361983 2241319 2210543 2207447

Loop of length 9:

1592822 1603185 1596873 1595290 1600827
1595730 1597020 1597694 1597871

Loop of length 13(!):

255998 260160 260451 259729 258099
257490 260364 262053 259389
258628 258775 257813 256619


The above loops are a complete set of all loops of length 5 or
greater which consist entirely of numbers under 5,000,000.


Another interesting result not posed as part of the puzzle...
The longest terminating chain found so far is the one starting
with the number 913669, which takes 128 steps to reach the prime
number 1552561."

***

On June 15, 2023 Giorgos Kalogeropoulos wrote:

Here are the missing loops of length 6 and higher up to n=5*10^8

Loop of length 6
25781970,25805244,25807866,25809472,25810027,25792359
33762788,33779454,33781166,33782481,33769476,33770781
45528862,45541032,45545057,45541118,45546829,45532799
47661878,49828340,49862551,47694591,47694418,47697355
149980470,150197866,150198641,149999471,149998244,150070839

Loop of length 7
6516882,6517170,6517450,6647811,6643710,6717542,6720549
6773510,6780597,6777330,6777953,6777369,6775605,6775039
7637348,7640754,7756539,7734785,7734001,7732082,7732827
59553638,59571357,59568777,59566725,59566481,59566248,59567959
95093178,95782287,95746465,95741897,95131919,95123230,95144073
283647264,283716033,283711505,283695756,283697570,283717720,283718205
445397284,445419618,445429025,445419853,445416946,445419296,445429809

Loop of length 8
10856832,10857440,10857696,10864379,10862530,10867924,10872907,10857455
306012300,306014348,310514580,311089625,311057271,306119841,306089490,306139797
399876358,399893546,399984596,399993267,399964665,399954208,399955313,399908339

Loop of length 9
19272214,19272550,19273163,19272896,19286024,19287457,19286905,19284689,19275839

Loop of length 10
92544058,92993406,94402413,94399820,94401274,94401776,94407125,93651853,93494907,92550497

Loop of length 12
416251858,417971931,417931805,417885662,417887997,417767058,418626678,418677137,418185455,418126358,418134589,416259323

Loop of length 23
4631278, 4687800, 4688433, 4678545, 4678252, 5847819, 5763042, 5763250, 5786320, 5786963, 5783577, 5721354, 5756685, 5756276, 5780730, 5781800, 5810725, 5805005, 4643999, 4633330, 4634608, 4636193, 4631807

***