Instance number u(i,j) is divisible with r=(n^k-1) just if corresponding class number g(i) is divisible with r. The number of all numbers that are not divisible with r is φ(r), where φ is Euler function.
If g(i) is not divisible with r, then its class must be the self class (not nested) and such class has k instances.
Therefore:
φ(n^k-1) mod k = 0
E-classes are classes satisfying:
gcd(g,r)=1
where gcd() denotes the greatest common divisor. For example φ(2^4-1) = φ(15) = 8 (see rows marked by asterisk).
G(2,4)
0
* 1 2 4 8
3 6 12 9
5 10
* 7 14 13 11
15
We see that 8 mod 4 = 0.
G-quotient g(n,k) = φ(n^k-1)/k
n \ k | 1 2 3 4 5 6 7 8 9 10
--------------------------------------------------------
1 | - - - - - - - - - -
2 | 1 1 2 2 6 6 18 16 48 60
3 | 1 2 4 8 22 48 156 320 1008
4 | 1 4 12 32 120 288 1512 4096
5 | 2 4 20 48 280 720 5580
6 | 4 12 56 216 1240
7 | 2 8 36 160
8 | 6 18 144
9 | 4 16 96
10 | 6 30
n \ k | 11 12 13 14 15 16 17 18 19
----------------------------------------------------
2 | 176 144 630 756 1800 2048 7710 7776 27594
Basic algebraic terms
All classes with gcd(g,r)=1, r=n^k-1 are self-classes.
System G(n,k) has n^k instances.
If p ε P, only n instances (instances from G(n,1)) are nested to
the system G(n,p). All others classes are self-classes and have
k transpositions. Therefore p\(n^p-n), i.e. n^p-n=0(mod p), i.e. the
Fermat theorem.
Fermat quotient f(n,k) = ((n^(k-1))-1)/k
k\n| 2 | 3
------------------------
3 | 1 | 1
5 | 3 | 16
7 | 9 | 104
11 | 93 | 5368
13 | 315 | 40880
17 | 3855 | 2532160
19 | 13797 | ...
23 | 182361| ...
Let us evaluate the sum of instance numbers in all classes of G(3,3):
0 0
1 3 9 13
2 6 18 26
4 12 10 26
5 15 19 39
7 21 11 39
8 24 20 52
13 13
14 16 22 52
17 25 23 65
26 26
Any of those sum is divisible by (n^k-1)/(n-1) = (3^3-1)/(3-1) = 13.
In G(n,k), it holds for each class g(i):
∑ u(i,j) = 0 [ mod (n^k- 1)/(n-1)]
j
Let the level L(g) of a class g in G(n,k) is the sum of particular
numbers (digits) in any instance of the class.
It holds:
L(g(i)) = k/q* ∑{j} u(i,j) / c(k,q)
where q is number of transpositions (i.e. order of the original nested system)
and c(k,q) is nesting quotient.
For example in G(3,3):
g| | L(g) |k/q| ∑ uij
--+------+-------+---+------
0| 000 | 0 | 3 | 0
1| 001 | 1 | 1 | 13
2| 002 | 2 | 1 | 26
4| 011 | 2 | 1 | 26
5| 012 | 3 | 1 | 39
7| 021 | 3 | 1 | 39
8| 022 | 4 | 1 | 52
13| 111 | 3 | 3 | 13
14| 112 | 4 | 1 | 52
17| 122 | 5 | 1 | 65
26| 222 | 6 | 3 | 26
...
L(7) = 3/3* 39/c(3,1) = 1*39/ ((3^3-1)/(3-1)) = 39/13 =3
L(8) = 3/3* 52/c(3,1) = 1*52/ ((3^3-1)/(3-1)) = 52/13 =4
...
L(13)= 3/1* 13/c(3,1) = 3*13/ ((3^3-1)/(3-1)) = 39/13 =3
...
Schematic algebra