CNB12 Formulae for d,e,f,g ; Table of Primorials
In this Blog we shall give formulae for obtaining the frequencies of the 2-vecs d, e, f, and g at the ends of calculations of the cap primorials 2 , 2Λ3 , 2Λ3Λ5 , … . We shall index them by k, where p(k) is the kth prime. We also give, first, a table of primorials for k = 1 to 8, together with values for what we call reduced primorials. We shall use the symbol X(k) (i.e. CHI-k) for the kth primorial. And X(k, -1), X(k, -2) will denote the first and second reduced primorials respectively, to be explained next.
The first three primorial formulae (on numbers, not cycle-numbers)
The following products are calculated by sequential ordinary multiplication (i.e. not using cap products):
Xk = pk p(k-1) p(k-2) … p3 p2 p1 ( = pk# )
Xk(-1) = (pk-1) (p(k-1)-1) … (p2-1) (p1-1) ( = (pk-1)# )
Xk(-2) = (pk-2) (p(k-1)-2) … (p2-2) (p1-2) ( = (pk-2)# )
The first three primorial formulae (on cycle-numbers)
When computing primorials on cycle-numbers, the same formulae apply, but with cap multiplication. For example,
X3 = 2’ Λ 3’ Λ 5’ = 30’
Table of the Primorials on numbers
| k | pk | Xk | Xk(-1) | Xk(-2) |
| 1 | 2 | 2 | 1 | 1 |
| 2 | 3 | 6 | 2 | 1 |
| 3 | 5 | 30 | 8 | 3 |
| 4 | 7 | 210 | 48 | 15 |
| 5 | 11 | 2310 | 480 | 135 |
| 6 | 13 | 30030 | 5760 | 1485 |
| 7 | 17 | 510510 | 92160 | 22275 |
Formulae for d, e, f, g
We shall now give formulae for the frequencies of 2-vecs d, e, f, g for two cases: (i) when the cap primorials are calculated from rows of the cycle-number matrix C, and (ii) when the cap primorials are calculated from a matrix called C’ which is derived from C in a manner to be described in a later blog.
Case (i) : calculations from C
d(k) = X(k-1) – X(k-1 ; -1) ;
e(k) = d(k).(p(k) – 1) ;
f(k) = g(k-1) ;
g(k) = g(k-1).(p(k) – 1) . [ Observe that: g(k)/f(k) = e(k)/d(k) = (p(k) – 1) . ]
Table of d, e, f, g vakues, for C and k = 1, 2, 3, 4, 5
|
k |
pk – 1 |
dk |
ek |
fk |
gk |
|
1 |
1 |
0 |
0 |
1 |
1 |
|
2 |
2 |
1 |
2 |
1 |
2 |
|
3 |
4 |
4 |
16 |
2 |
8 |
|
4 |
6 |
22 |
132 |
8 |
48 |
|
5 |
10 |
162 |
1620 |
48 |
480 |
Case (ii) : calculations from C’
d'(k) = 2(X(k) – X(k ; -2)) ;
e'(k) = d'(k).(p(k) – 2) ;
f”(k) = 2X(k-1 ; -2) = 2g'(k-1) ;
g'(k) = X(k ; -2) = g'(k-1).(p(k) – 2) . Observe that: g'(k)/f'(k) = e'(k)/d'(k) = (1/2). (p(k) – 2) .
Table of d, e, f, g values, for C’ and k = 1 to 5
|
k |
p’k – 1 |
d’k |
e’k |
f’k |
g’k |
|
1 |
0 |
0 |
0 |
0 |
1 |
|
2 |
1 |
2 |
1 |
2 |
1 |
|
3 |
3 |
10 |
15 |
2 |
3 |
|
4 |
5 |
54 |
135 |
6 |
15 |
|
5 |
9 |
390 |
1755 |
30 |
135 |
[to be continued, with explanations of C’ ]