Problems & Puzzles: Puzzles

Puzzle 640. A sequence of semiprimes. JM Bergot sent the following nice puzzle: Start with the semiprime 161=7*23, make 723 from 7*23 and so on: Levels=6 161=7*23 723=3*241 3241=7*463 7463=17*439 17439=35813 35813=59*607 59607=3*3*3*7*179 Not a semiprime STOP Can you start with some other semiprime and get more than six levels of semiprimes? As a matter of fact, I found the following larger sequence, starting with the semiprime 100462 Levels=11 100462=2*50231 250231=37*6763 376763=23*16381 2316381=3*772127 3772127=19*198533 19198533=3*6399511 36399511=5531*6581 55316581=19*2911399 192911399=3631*53129 363153129=3*121051043 3121051043=11*283731913 11283731913=3*19*10771*18379 Not a semiprime STOP. Q. Send your largest sequence of this type.

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Contributions came from Flavio Torasso, Jan van Delden, Hakan Summakoğlu, J. K. Andersen, Jahangeer Kholdi  & Farideh Firoozbakht.

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Flavio wrote:

My largest solution starts with 68279314 and stops with 867065337994599 (Levels=14). Note also this special case: two (Levels=12) sequences starting with 155514118 and 158452843, both ending at 341642110179947.

Kind regards

 

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Jan wrote:

In the example the number n=p*q with p<=q, semiprime. Next terms in the sequence are constructed by concatenation of the type p.q. There are no longer solutions for n<=2*10^6.

Only type q.p is allowed:

 

Levels=10

1715069

2089.821

696607.3

409769.17

999437.41

39209.2549

205607.1907

55569511.37

265921.20897

8864040299.3

36673729.2417=3*107*1142483777 NOT a semiprime STOP

 

Both p.q and q.p are allowed:

 

Levels=21

28907

211.137

3.70379

17.21787

573929.3

7.819899

3.2606633

83.392851

27797617.3

8821.31513

17.51890089

3.583963363

6827.524969

23302133.293

10416689.2237

46900897.2221

41961973.11177

1398732437059.3

7.1998189195799

459089.156828391

3.153029718942797

966157.3263475521=3^2*241*1553*97523*29411 NOT a semiprime STOP

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Hakan, Jens, Jahangeer & Farideh wrote:

Levels = 16
636542506 = 2*318271253
2318271253 = 4999*463747
4999463747 = 23*217367989
23217367989 = 3*7739122663
37739122663 = 11*3430829333
113430829333 = 20549*5520017
205495520017 = 31601*6502817
316016502817 = 179*1765455323
1791765455323 = 19*94303445017
1994303445017 = 13*153407957309
13153407957309 = 3*4384469319103
34384469319103 = 19469*1766113787
194691766113787 = 47*4142378002421
474142378002421 = 1187*399445979783
1187399445979783 = 43*27613940604181
4327613940604181 = 17*254565525917893
17254565525917893 = 3*105467*54533852693 Not a semiprime.

Jahangeer & Farideh added:

We can also define a(n) as the smallest semiprime m such that

its related chain has a length of n. For n=1, 2, ...,14 and n=16, a(n) are :

6, 38, 34, 15, 265, 161, 1126, 4891, 1253, 250231, 100462, 49869178, 234139657, 68279314,  ? ,  636542506.

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