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[PrimeNumbers] sigma(n-1)+sigma(n)+sigma(n+1)

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  • Jon Perry
    Jack wrote: Because sigma(x) is odd if and only if x is a square or twice a square. The density of x such that sigma(x) is odd == 0. [jp :
    Message 1 of 9 , Jul 2, 2002
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      Jack wrote:

      Because sigma(x) is odd if and only if x is a square or
      twice a square.

      The density of x such that sigma(x) is odd == 0.

      [jp : http://mathworld.wolfram.com/DivisorFunction.html (eqn28)


      Try finding n such that n divides sigma(n-1)+sigma(n)+sigma(n+1).
      Heuristics would indicate an infinite number of such n...

      2, 5, 7, 33, 18336, 19262, 38184, 54722, ...

      I've just submitted this to the EIS.

      ---

      From the link, the equations can be morphed into:

      (n-1).sigma(n-1) = 2 + i.phi(n-1)
      n.sigma(n) = 2 + j.phi(n)
      (n+1).sigma(n+1) = 2 + k.phi(n+1)

      n(n^2-1).sigma(n(n^2-1)) = [2 + i.phi(n-1)].[2 + j.phi(n)].[2 + k.phi(n+1)]

      so the answers occur whenever the RHS contains n^2 as a factor.

      Jon Perry
      perry@...
      http://www.users.globalnet.co.uk/~perry/maths
      BrainBench MVP for HTML and JavaScript
      http://www.brainbench.com


      -----Original Message-----
      From: Phil Carmody [mailto:thefatphil@...]
      Sent: 02 July 2002 13:35
      To: primenumbers
      Subject: RE: [PrimeNumbers] Factor patterns


      --- Jon Perry <perry@...> wrote:
      > A swerve.
      >
      > Can you explain why sigma(n-1)+sigma(n)+sigma(n+1) is nearly always
      > even?
      >
      > (PARI/GP)
      >
      > for
      > (n=2,1000,write("sigmax.txt",(sigma(n-1)+sigma(n)+sigma(n+1))%2))


      Look at sigma(n)%2.

      Mostly 0.

      Each of the many remotely isolated 1s will cause 3 consecutive 1s in
      your expression. 101 causes a 11011 in yours, and 11 causes 1001 in
      yours.

      No magic.

      Phil

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    • djbroadhurst
      ... I believe not; you can do it mod 6, mod 12, mod 24 and then there are exceptions. Easy to see why: we require a modulus M such that n^2=1 mod M for all n
      Message 2 of 9 , Jul 2, 2002
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        > Are there any more numbers sharing this property?
        I believe not; you can do it mod 6, mod 12, mod 24
        and then there are exceptions.
        Easy to see why: we require a modulus M such that
        n^2=1 mod M for all n with gcd(n,M)=1
        M=24 is easy.
        We cannot have any larger M with gcd(M,5)=1,since
        5^2-1=24.
        So try M=12*5.
        Now we have problems with
        7^2-1=48
        unless M=12*5*7.
        Now we have problems with
        11^2-1=120
        It's clear enough that we are are losing out,
        though I do not have a proof..
        David
      • Jon Perry
        (sorry about the complete tripe in previous post - the n.sigma(n) eqv. 2 mod phi(n) is only true for n=p, and n=4,6,22). -- Continuing Jack s work: for
        Message 3 of 9 , Jul 3, 2002
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          (sorry about the complete tripe in previous post - the n.sigma(n) eqv. 2 mod
          phi(n) is only true for n=p, and n=4,6,22).

          --

          Continuing Jack's work:

          for (n=2,100000,if ((sigma(n-1)+sigma(n)+sigma(n+1))%(n-1)==0,
          write("sigmaconn.txt",n,":",sigma(n-1)%(n-1),":",+sigma(n)%(n-1),":",sigma(n
          +1)%(n-1),":",(sigma(n-1)+sigma(n)+sigma(n+1)))))

          2,3,24,64,227,291,784,1883,7731,18547,25723,30397,94358

          for (n=2,100000,if ((sigma(n-1)+sigma(n)+sigma(n+1))%(n+1)==0,
          write("sigmaconp.txt",n,":",sigma(n-1)%(n+1),":",+sigma(n)%(n+1),":",sigma(n
          +1)%(n+1),":",(sigma(n-1)+sigma(n)+sigma(n+1)))))

          8,21,22,23,57,157,505,1053,2147,2273,3311,4679,5931,7898,22682

          --

          A related question - when does sigma(n,2)%sigma(n,1)==0?

          If n is a square, and also:

          20,50,117,180,200,242,325,450,468,500,578,605,650,800,968,980

          for (n=2,10000, if (sigma(n,2)%sigma(n,1)==0 &&
          !issquare(n),write("sigmasigmasq.txt",n)))

          Anyone spot the missing link?

          Jon Perry
          perry@...
          http://www.users.globalnet.co.uk/~perry/maths
          BrainBench MVP for HTML and JavaScript
          http://www.brainbench.com
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