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• ... 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
Message 1 of 9 , Jul 2, 2002
--- 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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• ... Are there any more numbers sharing this property? N.
Message 2 of 9 , Jul 2, 2002
> Likewise, if a*b=-1 mod 24 then a+b=0 mod 24
> with residue pairs {1,23},{5,19},{7,17},{11,13}
> since n^2=1 mod 24 for all n with gcd(n,24)=1

Are there any more numbers sharing this property?

N.
• 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 3 of 9 , Jul 2, 2002
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

--- 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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losing the ability to manage files on the user's own
hard disk." - Prof. Stuart E Madnick, MIT.
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• ... 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 4 of 9 , Jul 2, 2002
> 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
• (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 5 of 9 , Jul 3, 2002
(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)))