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.
*
* *
