definition of anti-divisor

[N. J. A. Sloane]

should be changed to:
Definition: If an odd number i in the range 1 < i <= n divides N
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Date: Sat, 21 Jul 2007 07:31:25 -0500
From: "Diana Mecum" <diana.mecum at gmail.com>
To: njas at research.att.com
Subject: Re: definition of anti-divisor
Cc: seqfan at ext.jussieu.fr
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Dr. Sloane,

As far as the second definition, and

> %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.

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

According to the second definition, above, an antidivisor must first be a
non-divisor of x. {4, 5, 6, 7} are not non-divisors of 3. Therefore, I
believe that the second definition is correct as it stands.

Diana

On 7/21/07, N. J. A. Sloane <njas at research.att.com> wrote:
>
>
> 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
> should be changed to:
>
> Definition: If an odd number i in the range 1 < i <= n divides N
> 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
>
>


-- 
"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.

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Dr. Sloane,<br><br>As far as the second definition, and <br><br>> %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.<br><br>> This definition also fails for n = 3: it gives 5 antidivisors,<br>2,4,5,6,7.<br><br>According to the second definition, above, an antidivisor must first be a non-divisor of x. {4, 5, 6, 7} are not non-divisors of 3. Therefore, I believe that the second definition is correct as it stands.
<br><br>Diana<br><br><div><span class="gmail_quote">On 7/21/07, <b class="gmail_sendername">N. J. A. Sloane</b> <<a href="mailto:njas at research.att.com">njas at research.att.com</a>> wrote:</span><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
<br>Dear seqfans, There are currently two versions<br>of the definition of anti-divisor in the OEIS:<br><br>%C A066272 If an odd number d in the range 1 < d < n divides N<br>where N is any one of 2n-1, 2n or 2n+1<br>
then N/d is called an anti-divisor of n.<br><br>%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.
<br><br>But this definition fails for n = 3, as someone mentioned last night.<br>We know from the OEIS that 3 has a single antidivisor, 2.<br>According to this definition 3 has no antidivisors.<br><br>There is also this program, which I have not checked:
<br>%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} ]
<br><br>The other definition is:<br><br>%I A130799<br>%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,<br>%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,<br>%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
<br>%N A130799 Triangle read by rows in which row n (n>=3) list the anti-divisors of n.<br>%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.
<br>%C A130799 There are no anti-divisors of 1 and 2.<br>%e A130799 Anti-divisors of 3 through 20:<br>%e A130799 3: 2<br>%e A130799 4: 3<br>%e A130799 5: 2, 3<br>%e A130799 6: 4<br>%e A130799 7: 2, 3, 5<br>%e A130799 8: 3, 5
<br>%e A130799 9: 2, 6<br><br>This definition also fails for n = 3: it gives 5 antidivisors,<br>2,4,5,6,7.<br><br>The term anti-divisor seems to be due to Jon Perry.<br>I wish I understood the motivation for the definition!
<br><br>There are links to various webpages of his, but they are all broken<br>and he has not responded to my emails.<br><br>It seems to me that both of the above definitions are incorrect, and<br>should be changed to:<br>
<br>Definition: If an odd number i in the range 1 < i <= n divides N<br>where N is any one of 2n-1, 2n or 2n+1<br>then d = N/i is called an anti-divisor of n.<br><br>Equivalently, an anti-divisor of n is a number d in the range [1..n]
<br>which does not divide n and is either an odd divisor of 2n-1 or 2n+1,<br>or an even divisor of 2n.<br><br>Now both definitions seem to work correctly for n=3, giving<br>a single anti-divisor, 2.<br><br>But I'm not too confident about all this - comments anyone?
<br>As I said, I wish I understood the motivation for the definition!<br><br>Neil<br><br></blockquote></div><br><br clear="all"><br>-- <br>"God made the integers, all else is the work of man." <br>L. Kronecker, Jahresber. DMV 2, S. 19.

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