Improved Formulations For Comp(z) and Prime(z)


by

Huen Y.K.

CAHRC, P.O.Box 1003, Singapore 911101
http://web.singnet.com.sg/~huens/
email: huens@mbox3.singnet.com.sg

(A short communication - 1st released: 16/9/97.)


Abstract

Although the original sequence algebraic generating function for the composite number sequence called Comp(z) is compact, it is algorithmically inefficient since it does not discriminate between odd and even integer divisors in compositeness testing [1,17,18,19]. Consequently the identity for Prime(z) = Nat(z) - 1/z - Normc(Comp(z)) is also algorithmically inefficient. Improvements can be made by recognising that Comp(z) is a compound sequence containing an even composite number sequence called Evencomp(z) and an odd composite number sequence called Oddcomp(z). In this paper the formulations for Evencomp(z) and Oddcomp(z) are derived directly without using Comp(z). Once these are obtained, the new identities for Comp(z) and Prime(z) can be expressed as follows: (i) Comp(z) = Normc(Oddcomp(z)+Evencomp(z)), and (ii) Prime(z) = (Odd(z) - 1/z) - Normc (Oddcomp(z)). These new formulations save much computing resources in processing by symbolic packages.


1. Derivation Of Oddcomp(z)

To derive directly the sequence algebraic generating function for the odd composite sequence called Oddcomp(z), the formulation shown in equation (1) will be used.

................................infinity
..................................-----
...................................\......................1
.......Oddcomp(z):=.......) -------------------------------- .....................................(1).
.................................../.........(2 i + 1).....(4 i + 2)
..................................----- z...............(z.............. - 1)
.................................i = 1

To explain what this equation does, the first three terms for i = 1 to 3 will be expanded using Maple V R 3 as shown from equations (2a) to (2c). It can be seen that for each value of i, the sequence generated contains odd composites only. If the equation of odd composites shown in equation (1) is expanded globally, we will get the global Oddcomp(z) sequence.

General Maple Program Line: Fi(z):=series(1/(z^(2*i+1)*(z^(2*(2*i+1))-1)),z=infinity,50)

i=1;

F1(z):=series(1/(z^(2*1+1)*(z^(2*(2*1+1))-1)),z=infinity,50);

.....................................1........1......1.......1.......1.......1.......1..........1
......................F1(z) := ---- + --- + --- + --- + --- + --- + --- + O(---)..........................(2a).
.......................................9.......15......21....27.....33.....39.....45........51
.....................................z........z........z.......z........z.......z........z..........z

i=2;

F2(z):=series(1/(z^(2*2+1)*(z^(2*(2*2+1))-1)),z=infinity,50);

.............................................1.......1......1.......1...........1
...............................F2(z) := --- + --- + --- + --- + O(---).....................................(2b).
..............................................15......25....35.....45.........55
.............................................z........z.......z.......z...........z

i=3;

F3(z):=series(1/(z^(2*3+1)*(z^(2*(2*3+1))-1)),z=infinity,50);

.................................................1.......1.......1...........1
...................................F3(z) := --- + --- + --- + O(---)........................................(2c).
..................................................21.....35.....49.........63
.................................................z.......z........z...........z

The Oddcomp(z) generating function will subsequently be used to derive a new formulation for Prime(z) (see section 4).


2. Indirect Derivation of Evencomp(z)


Evencomp(z) can be obtained from the difference between the normalised Comp(z) and the normalised Oddcomp(z) as shown in equation (3). This will be described first since this is a new formulation developed via the Oddcomp(z) route.

Evencomp(z) = Normc(Comp(z)) - Normc(Oddcomp(z)) ...................................(3).

A finite working example will be demonstrated in example 1:

Example 1:

Comp(z):=sort(series(sum(1/(z^i*(z^i-1)),i=2..40),z=infinity,40));

........................1........1........2.........2........1........2........4......2......2......3.......4.......4.......2.......2
Comp(z) := O(---) + ---- + ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + --- + --- + ---
.........................40.......4........6.........8........9.......10.....12.....14.....15....16.....18.....20.....21.....22
........................z........z.........z.........z.........z.........z........z.......z.......z......z........z.......z.......z.......z

..........6.......1.......2.......2......4.......6.......4.......2.......2.......2.......7......2......2
......+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- ......................(4).
...........24.....25.....26.....27.....28.....30.....32....33.....34.....35.....36....38....39
..........z.......z........z.......z........z.......z.......z.......z........z........z.......z.......z.......z

Normc(Comp(z)):=sort(1/z^4+1/z^6+1/z^8+1/z^9+1/z^10+1/z^12+1/z^14+1/z^15+1/z^16 +1/z^18+1/z^20+1/z^21+1/z^25+1/z^35+1/z^27+1/z^33+1/z^39+1/z^22+1/z^24+1/z^26+1 /z^28+1/z^30+1/z^32+1/z^34+1/z^36+1/z^38);

................................1.....1......1......1.......1......1.......1.......1.......1.......1.......1.......1.......1......1
Normc(Comp(z)):= ---+ ---+ ---+ ---+ --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
.................................4.....6.......8......9......10....12.....14.....15.....16.....18.....20.....21.....22.....24
................................z.....z.......z......z........z......z.......z........z........z........z........z........z........z......z

..........1........1......1.......1.......1.......1.......1.......1.......1.......1.......1.......1
.......+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- ..........................(5).
...........25......26....27.....28.....30.....32.....33.....34.....35.....36.....38.....39
...........z........z......z.......z........z.......z........z.......z........z.......z.........z.......z

Oddcomp(z):=series(sum(1/(z^(2*i+1)*(z^(2*(2*i+1))-1)),i=1..40),z=infinity,40);

.....................................1.......2.......2.......1.......2.......2.......2.......2..........1
...........Oddcomp(z) := ---- + --- + --- + --- + --- + --- + --- + --- + O(---) ................(6).
........................................9......15......21.....25.....27....33.....35.....39........45
.....................................z........z........z........z.......z.......z.......z........z.........z

Normc(Oddcomp(z)):=1/z^9+1/z^15+1/z^21+1/z^25+1/z^27+1/z^33+1/z^35+1/z^39;

..............................................1.......1.......1.......1.......1.......1.......1.......1
.......Normc(Oddcomp(z)) := ---- + --- + --- + --- + --- + --- + --- + --- .......................(7).
................................................9......15......21.....25.....27.....33.....35...39
..............................................z.......z........z........z........z........z........z.......z

Evencomp(z):=sort(Normc(Comp(z))-Normc(Oddcomp(z));

............................1........1.......1.......1.......1.......1.......1.......1.......1.......1......1......1.......1......1
Evencomp(z) := ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
.............................4........6.......8.......10......12.....14.....16....18....20.....22.....24.....26....28....30
............................z........z.......z.......z.........z.......z........z.......z.......z.......z........z.......z.......z.......z

..........1......1.......1.........1
......+ --- + --- + --- + --- .............................................................................(8).
...........32.....34.....36......38
..........z.......z.......z.........z


3. Direct Derivation of Evencomp(z)


Evencomp(z) can also be derived directly just like Oddcomp(z) in equation (1). The equation of even composites is given by equation (9) as follows:

Evencomp(z):=series(sum(1/(z^(2*i)*(z^(2*(2*i-1))-1)),i=1..40),z=infinity,40);

Evencomp(z):=

...1.......1........1........2.......1......1.......3.......1.......1.......3.......1.......2......3......1......1......3
---- + ---- + ---- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
.....4.......6........8.......10......12....14.....16....18.....20.....22....24.....26.....28....30....32....34
...z........z........z.........z.......z.......z........z.......z.......z........z.......z.......z.......z......z.......z......z

.............3......1...........1
........+ --- + --- + O(---) ..............................................................................(9).

.............36....38..........40
............z......z............z
Note that both Evencomp(z) and Oddcomp(z) are in unnormalised forms. To normalise them, one needs the Normc( ) operator.

4. New Formulation For Prime(z)


An alternative identity for Prime(z) is derived from the difference between Odd(z) and Oddcomp(z) as shown in equation (10):

Prime(z) = (Normc(Odd(z)) - 1/z) - Normc (Oddcomp(z)).........................................(10).

Example 2 will be used to demonstrate the computations of a finite sequence of Prime(z):

Example 2:

.......................................1.......1.......1.......1.......1.......1.......1........1
Normc(Oddcomp(z)) := ---- + --- + --- + --- + --- + --- + --- + --- ..........................(11).
.........................................9.......15......21.....25.....27.....33.....35.....39
.......................................z........z.......z........z........z........z.......z........z

.........................................1.......1.......1......1........1......1.......1......1.......1.......1.......1......1.......1
Normc(Odd(z))-1/z := O(---) + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
.........................................41........3.......5......7........9......11....13....15.....17.....19.....21....23.....25
........................................z.........z........z......z.........z......z.......z.......z.......z........z.......z.......z.......z

. ..........1......1......1.......1.......1.......1.......1
.......+ --- + --- + --- + --- + --- + --- + --- ...........................................................(12).
.............27....29....31.....33.....35.....37.....39
............z......z.......z........z........z........z.......z

.......................1........1........1.......1........1........1.......1.......1.......1......1.......1.......1
Prime(z) = O(---) + ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + --- + --- ...........(13).
.........................41......3........5........7.......11......13....17.....19......23.....29.....31.....37
.......................z.........z........z........z........z.........z.......z.......z........z........z.......z.......z

This is the new formulation for Prime(z) which does not depend on Comp(z).


5. Conclusions


The direct derivations of Oddcomp(z) and Evencomp(z) are interesting but we still cannot bypass the use of the operator Normc( ). This does not imply that there is something wrong with the Normc( ) operator. It is simply that most symbolic packages do not at present support the Normc( ) function which prompted the author to look for ways to work around it. This obviously is impossible. From past experience, the author is of the opinion that the function Normc( ) is absolutely essential in sequence algebraic manipulations. We either await the inclusion of this function into commercial symbolic softwares or we could a develop a temporary useable software for sequence algebraic operations in the interim period. Since the author's work is more analytical than numeric, the present lack of Normc( ) functionality is just a minor nuisance. It does not impede dialactic research in sequence algebra. But the new formulations for Comp(z) and Prime(z) will definitely improve computational efficiency.

One another note, it is worthwhile mentioning that sequence algebraic formulations are inherently parallel and are most suitable for massively parallelled computers. This seems to be the trend of development in hardwares into the 21st centuary. The author predicts that it will be only a matter of time before sequence algebraic algorithms find favour with the computing communities when massively paralleled computers become commercially viable. By this the author has in mind not just 2^16 but 2^32 of the lowly 2-bit or 4-bit processors.


6. References


The papers included here are considered most likely to be relevant to the above topic without giving specifice references. These are the papers one should read if jargons and algebraic methods used in this paper are not familiar to readers.


1. Huen Y.K.: A Simple Introduction To Sequence Algebra, URL site: http://web.singnet.com.sg/~huens/

2. Huen Y.K.: Unsolved Problems In Sequence Algebra, URL site: http://web.singnet.com.sg /~huens/ .

3. Huen Y.K. : Methods Of Developing Sequence Algebraic Formulations For Comp(z) and Prime(z). URL site: http://web.singnet.com.sg /~huens/ .

4. Huen Y.K. : Lemmata, Corollaries, And Theorems In Sequence Order Analysis. URL site: http://web.singnet.com.sg/~huens/ .

5. Huen Y.K.: A Matrix Map for Prime and Non-prime Numbers, INT. J. Math. Educ. Sci. Technol., 1994, VOL. 25, NO.6, pp 913-920.

6. Huen Y.K.: Some Interesing Properties Of The Natural Number System, Int. J. Math. Educ. Sci. Technol., 1996, VOL.27, NO. 5, 685-691.

7. Huen Y.K.: Visual algebra and its applications, INT. J. Math. Educ. Sci. Technol.,1997, VOL.28 NO.3, 333-344.

8. Huen Y.K.: Is Pie Periodic? : INT. J. Math. Educ. Sci. Technol., 1997, VOL??,NO.?,???- ???,(In the press as proof paper mes100495).

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