S-cycles 1.

 

We can discover interesting relationships if we examine the sequences consisting of the smallest and largest divisors of natural numbers.

By definition, we should not think of trivial divisors here, because then we would get the two most trivial sequences: 1,1,1,1,1... or 1,2,3,4,5,... So the so-called it is worth thinking about your own (or internal) dividers.

Fix the definitions: a natural number a is a non-trivial divisor of a natural number n if and only if 1<a<n and there exists a number b such that a*b=n (in this case it is unnecessary to stipulate separately that b is also a natural number, this necessarily follows from the given conditions).

We would think that, based on the definition, we can now easily assemble the two sequences mentioned at the beginning. But - literally - right away - we are faced with a dilemma: what if a number does not have its own divisor? It seems logical that then the member of the sequences should be 0. (From now on, we will use the OEIS notation method, i.e. the nth member of the sequences will be denoted by a(n).)

In summary: the nth member of the SND sequence is the smallest proper divisor of n, if such exists, or 0, if such does not exist (SND: smallest nontrivial divisors).

On the other hand: the nth member of the LND sequence has the largest proper divisor of n, if such exists, or 0, if such does not exist (LND: largest nontrivial divisors).

Specifically, these sequences look like this:

 

n	SND(n)	LND(n)
1	0	0
2	0	0
3	0	0
4	2	2
5	0	0
6	2	3
7	0	0
8	2	4
9	3	3
10	2	5
11	0	0
12	2	6
13	0	0
14	2	7
15	3	5
16	2	8
17	0	0
18	2	9
19	0	0
20	2	10
21	3	7
22	2	11
23	0	0
24	2	12
25	5	5
26	2	13
27	3	9
28	2	14
29	0	0
30	2	15
31	0	0
32	2	16
33	3	11
34	2	17
35	5	7
36	2	18
37	0	0
38	2	19
39	3	13
40	2	20
41	0	0
42	2	21
43	0	0
44	2	22
45	3	15
46	2	23
47	0	0
48	2	24
49	7	7
50	2	25
51	3	17
52	2	26
53	0	0
54	2	27
55	5	11
56	2	28
57	3	19
58	2	29
59	0	0
60	2	30
61	0	0
62	2	31
63	3	21
64	2	32
65	5	13
66	2	33
67	0	0
68	2	34
69	3	23
70	2	35
71	0	0
72	2	36
73	0	0
74	2	37
75	3	25
76	2	38
77	7	11
78	2	39
79	0	0
80	2	40
81	3	27
82	2	41
83	0	0
84	2	42
85	5	17
86	2	43
87	3	29
88	2	44
89	0	0
90	2	45
91	7	13
92	2	46
93	3	31

 

At first glance, the two sequences are quite similar, but growing differences quickly become apparent. What can be said about them.

First, the coincidence of zeros: If SND(n)=0, then LND=0 and vice versa. This is a trivial consequence of the sequence formula.

It can also be observed that the non-zero terms of SND are always prime (this simply follows from the definitions of "smallest" and "prime"). In contrast, non-zero values of LND are completely mixed. We can simply state that LND(n) is prime if and only if n is the product of two primes. E.g. LND(35)=7, (SND(35)=5). A special case is if n is the square of a prime, then (and only then) SND and LND have the same value (e.g. SND(9)=LND(9)=3).

Another nice, and perhaps useful, but not particularly surprising connection is that if the values are not zero

SND(n)*LND(n)=n

The case can be concluded with this, but it cannot be denied that something small here encourages further thinking.

What can we do, we continue it.

 

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