## CNB12 Formulae for d,e,f,g ; Table of Primorials

September 16th, 2012

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 22Λ3 , 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,

X =  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’ ]