## Saturday, 31 July 2010

### ADDING HELP TO OEIS SEQUENCES

As I´m working on more than one small projects of programming sequences from The On-Line Encyclopedia of Integer Sequences, and as I always try to document my code the best I can: I like to add comments and help information to it, but after many hours of “copy and paste”, and being aware that when you are trying to do something with a computer: if you feel that everything is repetitive and boring, then it is for sure, that you are using the wrong procedure, so then, I decided to create a very simple tools to make my life easier.

The PHP code of these two on-line applications is almost the same than the one of OEIS2BibTeX, but this time changed to provide the help code for PARI/GP or Mathematica OEIS sequences functions.

Both applications can be used in two different ways:

* Entering the Sequence Id number with a HTML form: in a POST METHOD, or
* With the Id Number supplied to the PHP code within the link, using ?sequence=valid_ID

1) PARI/GP

1.1) HTML POST Method:
http://oeis2bibtex.netai.net/helpPARI_GP/

1.2) PHP Parameter:
http://oeis2bibtex.netai.net/helpPARI_GP/OEIS-PARI_GP-Help.php?sequence=A000200
Here you can change A000200 for the desired Sequence Id Number:

2) WOLFRAM MATHEMATICA

2.1) HTML POST Method:
http://oeis2bibtex.netai.net/helpMathematica/

2.2) PHP Parameter:
http://oeis2bibtex.netai.net/helpMathematica/OEIS-Mathematica-Help.php?sequence=A000200

2.3) Mathematica Code using Import:

As Mathematica can access on-line data, these functions can do the job too inside any Mathematica Notebook or Package:
OEISSequenceDescription[seq_String]:=Module[{dataloaded = StringJoin[Import["http://oeis.org/classic/?q=id%3a" <> seq <> "&fmt=3","Data"]], first, last},first = Flatten[StringPosition[dataloaded, "%N"], 1][[2]];last = Select[StringPosition[dataloaded, "%"][[All, 1]], # > first&][[1]];StringReplace[StringTake[dataloaded, {first + 10, last - 1}], "  " ~~ _ -> ""]]OEISAddHelp[seq_String]:=ToExpression[StringJoin[seq, "::usage=\"",seq, ": ", OEISSequenceDescription[seq], "\""]]

Archives:

References:

[2]-http://formatmysourcecode.blogspot.com/
[3]-Wolfram Mathematica Documentation Center: Import
[4]-E.Pérez Herrero-OEIS Utilities Page@ OEIS Wiki

## Sunday, 11 July 2010

### TOTIENT CARNIVAL

Euler's totient function: $latex \phi(n)$ is defined as the number of positive integers less than or equal to $latex n$, that are coprime to $latex n$, and using Iverson bracket, $latex \phi$ can be written as: (See reference [1])

$latex \phi(n) =\sum_{i=1}^{n}{[gcd(i,n)=1]}$

MULTIPLICATIVE BUT NOT COMPLETELY MULTIPLICATIVE FUNCTIONS:

The above summatory can be expressed with the aid of non completely multiplicative arithmetical funcions, using the fact that, some functions hold for inequalities similar to this one:

$latex f(n \cdot m) < f(n) \cdot f(m);$ if $latex gcd(n,m)\neq 1$ And that also hold for the properties that any multiplicative function has: $latex f(n)>0$

$latex f(n \cdot m) = f(n) \cdot f(m);$ if $latex gcd(n,m)= 1$

Then:

$latex \phi(n)=\sum_{i=1}^{n}{\bigg\lfloor \frac{f(i \cdot n)}{f(i)\cdot \f(n)}\bigg\rfloor}$

And applying this idea to arithmetical functions of common use, we can give, for the divisor sigma functions, the Piltz divisor functions and the squarefree kernel [4], respectively:

$latex \phi(n)=\sum_{i=1}^{n}{\bigg\lfloor \frac{\sigma_{k}(i \cdot n)}{\sigma_{k}(i)\cdot \sigma_{k}(n)}\bigg\rfloor}$

$latex \phi(n)=\sum_{i=1}^{n}{\bigg\lfloor \frac{\tau_{k}(i \cdot n)}{\tau_{k}(i)\cdot \tau_{k}(n)}\bigg\rfloor}$

$latex \phi(n)=\sum_{i=1}^{n}{\bigg\lfloor \frac{rad(i \cdot n)}{rad(i)\cdot rad(n)}\bigg\rfloor}$

Or we can construct an implicit formula for the Totient function:

$latex \phi(n)=\sum_{i=1}^{n}{\bigg\lfloor \frac {\phi(i)\cdot \phi(n)}{\phi(i \cdot n)}\bigg\rfloor}$

But we must take care that now:

$latex f(n \cdot m) > f(n) \cdot f(m);$ if $latex gcd(n,m)\neq 1$

so we must invert the fraction inside the floor function.

The same thing can be done with additive but not completely additive functions:

$latex \phi(n)=\sum_{i=1}^{n}{\bigg\lfloor \frac{\omega(i \cdot n)}{\omega(i) + \omega(n)}\bigg\rfloor} \; (n>1)$

OTHER FORMULAS:

This way of constructing new formulas, seems to be trivial and useless, but this is only an excuse to show how works the characteristic or indicator functions, that are underneath the behaviour of the arithmetical functions, and thus the basic Set Theory inside Number Theory.

And of course we can extend this collection of formulas to other functions distinct than $latex \phi$, like for example $latex \pi$ the Prime Counting Function:

$latex \pi(n)=\sum_{i=2}^{n}{\bigg\lfloor \frac{i+1}{\sigma_{1}(i)}\bigg\rfloor} \; (n>1)$

$latex \pi(n)=\sum_{i=2}^{n}{\bigg\lfloor \frac{\phi(i)}{i-1}\bigg\rfloor} \; (n>1)$

$latex \pi(n)=\sum_{i=2}^{n}{\bigg\lfloor \frac{2}{\tau(i)}\bigg\rfloor}=\sum_{i=2}^{n}{\bigg\lfloor \frac{k}{\tau_{k}(i)}\bigg\rfloor} \; (n>1)$

Archives:

References:

[1]-Peter Luschny - OEIS-Wiki: Sequences related to Euler's totient function.
[2]- N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences.
A000720: pi(n), the number of primes <= n. Sometimes called PrimePi(n) to distinguish it from the number 3.14159... [3]- N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences.
A000010: Euler totient function phi(n): count numbers <= n and prime to n. [4]- R. Muller, The On-Line Encyclopedia of Integer Sequences.
A007947: Largest squarefree number dividing n (the squarefree kernel of n)