Question related to sequence A066452

[Joshua Zucker]

I don't understand it either, but at least I could use the wayback
machine to track down the URL given, at
  http://tinyurl.com/2xuvgr

And it says
  The anti-phi function is defined as the numbers <n that do not have
any anti-divisor as a factor.

Which may or may not be what they actually mean ... but at least it's
another possible interpretation to try.

--Joshua Zucker


On 6/30/07, Diana Mecum <diana.mecum at gmail.com> wrote:
> Folks,
>
> I am trying to add extension terms for sequence A066452. I have a question.
>
> This is the internal text describing the sequence:
>
> %I A066452
> %S A066452
> 1,1,2,1,4,1,4,4,3,2,8,3,7,7,9,2,8,5,10,10,8,6,19,6,12,9,9,8,22,9,12,
> %T A066452
> 12,15,10,31,9,11,14,24,13,23,9,24,17,16,10,35,15,23
> %N A066452 Anti-phi(n).
> %H A066452 Jon Perry, <a
> href="http://www.users.globalnet.co.uk/~perry/maths/antidivisorother2.htm
> ">
>                Anti-phi function</a>
> %F A066452 anti-phi(n) = number of integers <= n that are coprime to the
> anti-divisors of n
> %e A066452 10 has the anti-divisors 3,4,7. Therefore numbers coprime to
> 3,4,7 and less than
>                10 are are 1,2,5, therefore anti-phi(10)=3.
> %Y A066452 Cf. A058838, A066241.
> %Y A066452 Sequence in context: A024994 A051953 A079277 this_sequence
> A007104 A102627 A088296
> %Y A066452 Adjacent sequences: A066449 A066450 A066451 this_sequence A066453
> A066454 A066455
> %K A066452 nonn,more,easy
> %O A066452 2,3
> %A A066452 Jon Perry (perry(AT)globalnet.co.uk), Dec 29 2001
>
> I found a definition for "anti-divisor" as follows:
>
> "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."
>
> I see how Jon gets 3, 4 and 7 as anti-divisors of 10. However, 2 is not
> coprime to the anti-divisors of 10. He has 1, 2, and 5 as on the anti-phi
> list.
>
> The sequence which I derived for this sequence is:
>
> 1, 1, 2, 1, 4, 1, 4, 4, 3, 2, 2, 5, 3, 5, 4, 9, 2, 4, 5, 6, 6, 6, 6, 10, 5,
> 8, 6, 5, 8, 8, 9, 12,
>  7, 10, 7, 12, 9, 8, 9, 13, 13, 9, 9, 14, 10
>
> Can someone tell me if I am misunderstanding the definition of the sequence,
> or if I have found an error?
>
> Thanks,
>
> Diana M.
>
> --
> "God made the integers, all else is the work of man."
> L. Kronecker, Jahresber. DMV 2, S. 19.