A062775, related to pythagorean triples mod n / enhanced Solution

[Gottfried Helms]

Hi Seqfans -


An even more concise notation for my assumption of the
generation rules is the following definition:



Definition of A(i): ===================================================

    1) For the most entries, A(i) is just i^2. The "irregular" entries are
       integer multiples only of i, but not of i^2. So with an arbitrary
       index i, written in its decomposition into powers of primefactors,

       [1.1]  i = p^j * q^k * ...

       the corresponding entry of A may be written, using the new
       function g(p,j):


       [1.2]     A(i) = i *   g(p,j) * g(q,k) * ...

                where g is a function defined below and is applied to all
       prime-powers of i.
       If the exponent of a prime-power is 1, the function results just that
       prime.

       This new notation has the advantage, that all calculation can be done
       in integer mode.

Definition of g(p,j): ==================================================

   2.1) if j=1, that means that prime-power is squarefree. This specific
      short-definition for j=1 is not really needed for p=odd prime,
      since it would result from the general formula, [2.2] too:

      [2.1]  if j=1      g(p,j) = p



   2.2) for primes p>2 ; if j=1 then the function returns just p

                                                    (j-2)/2
                                                /  p          if j even
                                    j    j-1    |
      [2.2] if p>2       g(p,j) = p   + p    -  |
                                                |   (j-1)/2
                                                \  p           if j odd




   2.3) for p=2 a small correction is needed; the two rightmosts terms have
      to be multipied by 2:
                                                    (j-2)/2
                                                /  p          if j even
                                 j       j-1    |
      [2.3] if p=2     g(p,j) = p   + 2*p    -2*|
                                                |   (j-1)/2
                                                \  p           if j odd

       in shorter form:

                                                    j/2
                                                /  2          if j even
                                 j+1            |
      [2.4]            g(2,j) = 2            -  |
                                                |   (j+1)/2
                                                \  2           if j odd

------------------------------------------------------------------------


It's worth to be noted, that this notation reflects also the mentioned
"multiplicative"-ness of the sequence, but explaines for the cases,
where multiplication is erroeneous.

 > ||> Comments:  a(n) is multiplicative and for a prime p: a(p) = p^2.


For instance

     A(15) = A(3*5) = A(3) * A(5)

but

     A(12) <> A(2)*A(6)

This multiplicativeness A(i*j) = A(i)*A(j) is only valid, if gcd(i,j)=1.

The reason is, that the common primefactors of i and j have to be
collected to a common set of powers of primes, and then the multiplica-
tiveness can be applied

So, for instance the example

     A(12) = A(2*6) <> A(2)*A(6)     is not multiplicative

but
     A(12) = A(4*3) = A(4)*A(3)    is.

This can easily be seen, if A(i*j), A(i),A(j) are written in the
form with the g-function and as the product of their decompositions of
powers of their prime-factors.


Gottfried Helms

----------------------------------------------------------

Appendix: reference for the sequence A

> ---------------------------------
> ||> ID Number: A062775
> ||> URL:       http://www.research.att.com/projects/OEIS?Anum=A062775
> ||>
> ||> Sequence:  1,4,9,24,25,36,49,96,99,100,121,216,169,196,225,448,289,
> ||>            396,361,600,441,484, 529,864,725,676,891,1176,841,900,961,
> ||>            1792,1089,1156, 
> 1225,2376,1369,1444,1521,2400,1681,1764,1849,
> ||>            2904,2475,2116,2209,4032,2695,2900
> ||>
> ||> Name:      Number of Pythagorean triples mod n; i.e. number of 
> non-congruent
> ||>            solutions to x^2 + y^2 = z^2 mod n.
> ||>
> ||> Comments:  a(n) is multiplicative and for a prime p: a(p) = p^2.
> ||>
> ||>The starting number of this series is 1.
> 
> Anonymous wrote:
> 
>  > I've noticed that when the moebius function of n is either -1 or 
> +1,then the
>  > series term is n^2.  However when the moebius(n) = 0,then the 
> interesting set
>  > of numbers that I've given results.