Concatenated anti-divisors

[Max Alekseyev]

There are glitches in both Jonathan's definition of anti-divisor and
the quoted comment to A066272.

Namely, Jonathan forgot to mention that k < x, and in the comment to
A066272 it should be "1 < d <= n" not "1 < d < n".

After such corrections these two definitions become equivalent and
consistent with the numerical values of A066272.

Max

On 7/20/07, Maximilian Hasler <maximilian.hasler at gmail.com> wrote:
> It appears to me that the definition
>
> %C A066272 If an odd number d in the range 1 < d < n divides N where N
> is any one of 2n-1, 2n or 2n+1 then N/d is called an anti-divisor of
> n.
>
> is not equivalent to your definition:
>
> > Non-divisor: a number k which does not divide a given
> > number x. Anti-divisor: a non-divisor k of x with the
> > property that k is an odd divisor of 2x-1 or 2x+1, or
> > an even divisor of 2x.
>
> According to the first definition, n=3 cannot have an anti-divisor
> since there is no odd d, 1<d<n.




Dear seqfans, There are currently two versions
of the definition of anti-divisor in the OEIS:

%C A066272 If an odd number d in the range 1 < d < n divides N
where N is any one of 2n-1, 2n or 2n+1
then N/d is called an anti-divisor of n.

%e A066272 For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd divisors > 1 {5,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the anti-divisors of 18 are 4, 5, 7, 12. Therefore a(18) = 4.

But this definition fails for n = 3, as someone mentioned last night.
We know from the OEIS that 3 has a single antidivisor, 2. 
According to this definition 3 has no antidivisors.

There is also this program, which I have not checked:
%t A066272 antid[ n_ ] := Select[ Union[ Join[ Select[ Divisors[ 2n - 1 ], OddQ[ # ] && # != 1 & ], Select[ Divisors[ 2n + 1 ], OddQ[ # ] && # != 1 & ], 2n/Select[ Divisors[ 2*n ], OddQ[ # ] && # != 1 & ] ] ] }, # < n & ] ]; Table[ Length[ antid[ n ] ], {n, 1, 100} ]

The other definition is:

%I A130799
%S A130799 2,3,2,3,4,2,3,5,3,5,2,6,3,4,7,2,3,7,5,8,2,3,5,9,3,4,9,2,6,10,3,11,
%T A130799 2,3,5,7,11,4,5,7,12,2,3,13,3,8,13,2,6,14,3,4,5,9,15,2,3,5,9,15,7,
%U A130799 16,2,3,7,10,17,3,4,17,2,5,6,11,18,3,5,8,11,19,2,3,19,4,12,20,2,3,7
%N A130799 Triangle read by rows in which row n (n>=3) list the anti-divisors of n.
%C A130799 Non-divisor: a number k which does not divide a given number x. Anti-divisor: a non-divisor k of x with the property that k is an odd divisor of 2x-1 or 2x+1, or an even divisor of 2x.
%C A130799 There are no anti-divisors of 1 and 2.
%e A130799 Anti-divisors of 3 through 20:
%e A130799 3: 2
%e A130799 4: 3
%e A130799 5: 2, 3
%e A130799 6: 4
%e A130799 7: 2, 3, 5
%e A130799 8: 3, 5
%e A130799 9: 2, 6

This definition also fails for n = 3: it gives 5 antidivisors,
2,4,5,6,7.

The term anti-divisor seems to be due to Jon Perry.  
I wish I understood the motivation for the definition!

There are links to various webpages of his, but they are all broken
and he has not responded to my emails.

It seems to me that both of the above definitions are incorrect, and

where N is any one of 2n-1, 2n or 2n+1
then d = N/i is called an anti-divisor of n.

Equivalently, an anti-divisor of n is a number d in the range [1..n]
which does not divide n and is either an odd divisor of 2n-1 or 2n+1,
or an even divisor of 2n.

Now both definitions seem to work correctly for n=3, giving
a single anti-divisor, 2.

But I'm not too confident about all this - comments anyone?
As I said, I wish I understood the motivation for the definition!

Neil