# [seqfan] Re: a(n) | n

Antti Karttunen antti.karttunen at gmail.com
Sat Jan 27 06:36:18 CET 2018

```> ------------------------------
>
> Message: 16
> Date: Thu, 25 Jan 2018 14:36:56 -0500
> From: Frank Adams-Watters <franktaw at netscape.net>
> To: seqfan at list.seqfan.eu
> Subject: [seqfan] a(n) | n
> Message-ID: <1612ed2b661-1725-a51da at webjas-vae062.srv.aolmail.net>
> Content-Type: text/plain; charset=utf-8
>
> I've been looking recently at sequences where a(n) divides n for every n > 0.
>
> These come naturally in pairs, whose element-wise product is n.
>
> Here's a list, which is by no means complete:
>
> A000012 * A000027
> A000034 * A026741
> A000265 * A006519
> A003557 * A007947
> A006530 * A052126
> A007913 * A008833
> A020639 * A032742
> A028233 * A028234
> A033676 * A033677
> A034699 * A284600
> A060789 * A089128
> A065330 * A065331
> A060775 * A140271
> A122377 * A298734

Note that in many cases the iterates of at least the other member of
such a pair converge to 1 (or to a prime, or a squarefree number), and
counting the number of iterations to reach 1 (or a prime, or a
squarefree number) should give yet another sequence with some natural
meaning.

E.g. for A003557 * A007947, where A003557 = "n divided by largest
squarefree divisor of n" we have
A051903 (Maximal exponent in prime factorization of n) definable as
a(1) = 0; for n > 1, a(n) = 1 + a(A003557(n)).
Defining a similar recurrence with A032742 gives A001222, and ditto
with A052126, while using A028234 gives A001221.
Of course using some other function than 1+ at each step is possible.

Any non-trivial or even surprising examples?

Best regards,

Antti

>
> (A298734 I have just submitted.)
>
> It seems to me that this should be documented somewhere in the OEIS, but I'm not sure where. (Maybe in the index file somewhere?)
>
> Preserving this property is for me one good reason to have a(1) = 1 in sequences like A020439 (least prime factor) where a(1) is otherwise undefined.
>