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Re: Divisor Function

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  • Adam
    7203 is not a perfect square! Maybe you didn t mean if and only if. Part of if and only if is: sigma[2k](m)/sigma[k](m) is integer only if m is a square I
    Message 1 of 7 , Aug 21, 2007
      7203 is not a perfect square!

      Maybe you didn't mean "if and only if." Part of "if and only if" is:

      sigma[2k](m)/sigma[k](m) is integer only if m is a square

      I demonstrated by example: there exists a non square for which the
      fraction is an integer.

      --- In primenumbers@yahoogroups.com, Sebastian Martin <sebi_sebi@...>
      wrote:
      >
      > 7203 is NOT a perfect square
      >
      > The result is true only if only m is a perfect square: m=n^2
      >
      > m= 1, 4, 9, 16, 25, 36, ......
      >
      > Adam <a_math_guy@...> escribió:
      > --- In primenumbers@yahoogroups.com, Sebastian Martin
      <sebi_sebi@>
      > wrote:
      > >
      > > Hello all:
      > >
      > > I have obtained this result:
      > >
      > > DivisorSigma[2k,m]/DivisorSigma[k,m] is integer
      > > if and only if
      > > m is a perfect square
      > >
      > > DivisorSigma[k,m]=sum[d^k, d divisor of m]
      > >
      > > Can anyone prove this result?
      > >
      > > Sincerely
      > >
      > > Sebastian Martin Ruiz
      > >
      > >
      > > ---------------------------------
      > >
      > > Sé un Mejor Amante del Cine
      > > ¿Quieres saber cómo? ¡Deja que otras personas te ayuden!.
      > >
      > >
      > > [Non-text portions of this message have been removed]
      > >
      >
      > I obtain a lot of small counterexamples with k=1, some with k=2,
      and
      > k=3,m=6050 and k=4,m=7203. E.g., Maple output:
      >
      > > ifactor(7203);
      > 4
      > (3) (7)
      > > q1:=sigma[4](7203);
      > > q2:=sigma[8](7203);
      > > q2/q1;
      > q1 := 2726235765168410
      > q2 := 7247255655544865674860411008810
      > 2658337825414441
      >
      > All of my counterexamples "appear" to have square factors tho'.....
      >
      >
      >
      >
      >
      >
      > ---------------------------------
      >
      > ¡Descubre una nueva forma de obtener respuestas a tus preguntas!
      > Entra en Yahoo! Respuestas.
      >
      >
      > [Non-text portions of this message have been removed]
      >
    • Peter Kosinar
      ... As Adam already disproved the only if part, let s concentrate on the if one. First, there are two simple observations to make: 1) If p is a prime,
      Message 2 of 7 , Aug 21, 2007
        > DivisorSigma[2k,m]/DivisorSigma[k,m] is integer
        > if and only if
        > m is a perfect square
        >
        > DivisorSigma[k,m]=sum[d^k, d divisor of m]

        As Adam already disproved the "only if" part, let's concentrate on the
        "if" one. First, there are two simple observations to make:
        1) If p is a prime,
        DivisorSigma[k, p^e] = 1 + p^k + ... + p^ek = (p^[k(e+1)] - 1)/(p^k-1)
        2) If c and d are coprime,
        DivisorSigma[k, c*d] = DivisorSigma[k, c] * DivisorSigma[k, d]

        Thus, if m = p1^e2 * p2^e2 ... * pn^en is the prime factorization, we have
        DivisorSigma[k, m] =
        (p1^[k(e1+1)] - 1) / (p1^k - 1) *
        (p2^[k(e2+1)] - 1) / (p2^k - 1) *
        ...
        (pn^[k(en+1)] - 1) / (pn^k - 1).
        and DivisorSigma[2k, m] =
        (p1^[2k(e1+1)] - 1) / (p1^(2k) - 1) *
        (p2^[2k(e2+1)] - 1) / (p2^(2k) - 1) *
        ...
        (pn^[2k(en+1)] - 1) / (pn^(2k) - 1).

        The ratio of these two is ten equal to
        DivisorSigma[2k, m] / DivisorSigma[k, m] =
        (p1^[k(e1+1)] + 1) / (p1^k + 1) *
        (p2^[k(e2+1)] + 1) / (p2^k + 1) *
        ...
        (pn^[k(en+1)] + 1) / (pn^k + 1).

        Obviously, if all the exponents are even, each fraction of the product
        is an integer (the numerator factors algebraically in such case). This
        proves the "if" direction of your statement.

        The smallest counterexample to the "only if" part seems to be m=20, k=1:
        DivisorSigma[20, 2] = 1+4+16+25+100+400 = 546
        DivisorSigma[20, 1] = 1+2+4+5+10+20 = 42
        546 / 42 = 13

        Peter

        PS. Now I only have to hope that Yahoogroups do not mangle this mail
        beyond comprehensibility :-)

        --
        [Name] Peter Kosinar [Quote] 2B | ~2B = exp(i*PI) [ICQ] 134813278
      • Kermit Rose
        Sebastián Martín Ruiz Said: 1. Divisor Function Posted by: Sebastián Martín Ruiz sebi_sebi@yahoo.com sebi_sebi Date: Sun Aug 19, 2007 8:31 pm ((PDT))
        Message 3 of 7 , Aug 24, 2007
          Sebastián Martín Ruiz Said:


          1. Divisor Function
          Posted by: "Sebastián Martín Ruiz" sebi_sebi@... sebi_sebi
          Date: Sun Aug 19, 2007 8:31 pm ((PDT))

          Hello all:

          I have obtained this result:

          DivisorSigma[2k,m]/DivisorSigma[k,m] is integer
          if and only if
          m is a perfect square

          DivisorSigma[k,m]=sum[d^k, d divisor of m]

          Can anyone prove this result?

          Sincerely

          Sebastian Martin Ruiz



          Kermit Said:




          If m is a perfect square then

          m = p1^a1 p2^a2 p3^a3 . . . pj^aj

          where each of the a1, a2, .... aj are even.

          If d divides m then

          d = p1^b1 p2^b2 p3^b3 . . . pj^bj

          where each b1,b2,...bj are less than or equal to the corresponding
          a1,a2,...aj.

          The
          DivisorSigma[k,m]=sum[d^k, d divisor of m]

          is product of

          ( 1 + p1^k + p1^(2k) + p1^(3k) + . . . + p1^(a1*k) )
          (1 + p2^k + p2^(2k) + p2^(3k) + . . .+ p2^(a2*k) )
          . . .
          (1 + pj^k + pj^(2k) + pj^(3k) + . . . + pj^(aj * k) )


          =

          ( p1^( a1 * k) - 1) / (p1^k - 1)
          (p2^(a2 * k) - 1)/ ( p2^k - 1)
          . . .
          (pj^(aj*k) - 1) / (pj^k - 1)

          When will this product be divisible by

          ( p1^( a1 * k/2) - 1) / (p1^k - 1)
          (p2^(a2 * k/2) - 1)/ ( p2^k - 1)
          . . .
          (pj^(aj*k/2) - 1) / (pj^k - 1)

          ?

          To see the answer more readily,

          define a1 = 2 c1, a2 = 2 c2, ...aj = 2 cj

          Then we are asking when is

          [ p1^(2 c1 * k) -1 ) ( p2^(2 c2 * k) - 1 ) . . . ( 2 pj ^ ( 2 cj * k ) - 1) ]
          / [ p1^(c1*k) - 1) ( p2^(c2 * k) - 1 ) . . . ( 2 pj ^ ( cj * k ) - 1) ]


          = [ p1 ^ (c1 * k) + 1 ) ( p2 ^ (c2 * k) + 1 ) . . . (pj * ( cj * k) + 1)

          an integer?
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