# [seqfan] Re: New sequence?

Neil Sloane njasloane at gmail.com
Sat Apr 21 09:57:58 CEST 2012

```Richard, I entered your sequence - it is now A182256.
Note that A000048 has a b-file giving the first 100 terms.
Neil

On Fri, Apr 20, 2012 at 5:00 PM, Richard Guy <rkg at cpsc.ucalgary.ca> wrote:

> Well, that went over like a lead balloon!  Here's what I guess to be the
> next three members of A000048:  954437120, 1857283155, 3616814565.
>
> This gives  4096, 2, 4  for the next three members of the suggested
> sequence.  Conjecture: in this sequence, if  n  is 3-smooth. then
> a(n)  is a power of 2.  [probably easy to prove.]
>
> A000048  is a sort of pseudo-divisibility sequence.  E.g.
>
> 2 has ranks of apparition  4, 9 and 25 (and 49, ...???) in the
> sense that  A000048(n) is even just if  n  is a multiple of
> 4 or 9 or 25 (or 49 or ???)
>
> 5 has ranks  6, 13, 14, 17, 22, ???
>
> 7 has ranks  9, 13, 19, ??
>
> 11 has ranks 21, 25, 33, ??
>
> 13 has ranks 14, 18, 37, 38, ??
>
> 17 has ranks 10, 12, 21, 26, ??
>
> 31 has ranks 11, 25, ??
>
> However,  3  does not divide  A000048(15), so maybe I'm barking
> up the wrong tree.  Check?  Mod 2^15 + 1, there is a 2-cycle
> {10923,-10923}, a 6-cycle {3641,7282,14564,-3641,-7282,-**14564},
> three 10-cycles of which one is
> {2979,5958,11916,-8937,14895,-**2979,-5958,...}, and 1091
> 30-cycles.    R.
>
>
> On Thu, 19 Apr 2012, Richard Guy wrote:
>
>  Would an editor more competent than I like to enter the following
>> sequence into OEIS, if it's not there already (I'm not a good looker) ?
>>
>> [check & extend.  These are only hand calculations.  A000048 could
>> also easily be extended]  For  n = (0) 1 2 3 ...
>>
>> (0),0,0,2,0,2,4,2,0,8,4,2,16,**2,4,38,0,2,64,2,16,134,4,2,**256,32,4,
>> 512,16,2,1084,2,0,2054,4,159,
>>
>> It's the total length of all cycles which are strictly less than
>> the full length of  2n.
>>
>>   2^n  -  2 * n * A000048(n)
>>
>> a(2^k) = 0,  a(prime) = 2,  a(2p) = 4.
>>
>> There's a simple formula using the Moebius function (v. A000048).
>>
>> Let me know if I've made errors.   Thanks!   R.
>>
>>
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--
Dear Friends, I will soon be retiring from AT&T. New coordinates:

Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA