(Artikel ini sengaja saya tulis dalam bahasa Inggris, supaya dapat dibaca pula oleh mereka yang tidak bisa berbahasa Indonesia.)

We have discussed the 5*n* + 1 problem which has similar behavior to the 3*n* + 1 problem. Out of curiosity, I investigated what happens if we do 5*n* + 3 instead of 5*n* + 1, using a computer program made by Mr. Bee.

It turns out that some sequences ‘converge’ to 1 (as in the 5*n* + 1 problem), but some escape to infinity and the others are trapped in a loop. For examples, for *n*_{1} = 25, the sequence converges to 1; for *n*_{1} = 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 47, or 49 (and many others), the sequences run off to infinity; and for *n*_{1} = 43, 53, or 61 (and many others), the sequences are trapped in a loop. Below is the sequence obtained for *n*_{1} = 43.

My curiosity went further. Knowing the behavior of the sequences in the 5*n* + 3 problem, I would like to find out what happens if we do 5*n* + 1 and 5*n* + 3 alternately, every time we hit a number which is not divisible by 2 or 3. For example, for *n*_{1} = 5, the sequence converges to 1. The same also occurs for *n*_{1} = 43. However, if we start with *n*_{1} = 23, then we end up with a loop, as in the following table (and the next one).

Note that at Step 17, when we get 11, we do 5*n* + 3 and get *n*_{18} = 58 and *n*_{19} = 29. Next, we do 5*n* + 1 and get the following sequence (just add the step by 18), where we eventually go back to the previous Step 3.

It is interesting to observe what happens if we change the order of 5*n* + 1 and 5*n* +3 operations. For example, if we start with *n*_{1} = 11, with 5*n* + 1 operation first, then we will get a sequence which is convergent to 1; but if we apply 5*n* + 3 operation first, then we will get a loop.

Some loops are long, for example for *n*_{1} = 179, with 5*n* + 3 operation first.

The 5*n* + 1 vs 5*n* + 3 problem seems to have interesting behavior to investigate further. One question really bothers me: is there a sequence that runs off to infinity? If not, then the 5*n* + 3 operations seem to never win against the 5*n* + 1 operations; they always lose to or tie with the 5*n* + 1 operations. Is it really the case?

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Bandung, 05-09-2016