# [seqfan] Re: Pseudo-arithmetic progressions

Mon Jul 19 10:27:05 CEST 2010

```Remark. With respect to the majorizing sequence, I consider only cases when periods contain no two the same numbers.

Regards,

----- Original Message -----
From: Vladimir Shevelev <shevelev at bgu.ac.il>
Date: Sunday, July 18, 2010 21:56
Subject: [seqfan] Re: Pseudo-arithmetic progressions
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> Dear Seq Fans,
>
> Now I present the sequence minorizing all of such type sequences:
>
> %I A179541
> %S A179541 1,1,1,1,2,2,1,1,3,2,4
> %N A179541 a(n) is the least possible smallest period attainable
> by the action of a periodic sequence of binary operations
> <+>,<-> (see A179382,A179480), beginning with 2n-1<+>1
> or 2n-1<->1
> %C A179541 The minorizing sequence for all sequences of type
> A179382,A179480 with arbitrary perodic rotation of the binary
> operations <+>,<->.
> %e A179541 Let n=12, 2n-1=23. Considering periodic sequence
> <+>,<->,<+>,<->,..., we have 23<+>1=3, 23<-
> >3=5, 23<+>5=7, 23<->7=1, 23<+>1=3,... Thus
> a(12)<=4. It is not difficult to verify that a(12)>3. Thus
> a(12)=4.
> %Y A179541 A179382 A179480
> %K A179541 nonn
> %O A179541 2,5
>
> I think that calculations of new terms are rather difficult. I
> ask you to find more terms.
> With respect the corresponding majorizing sequence, I get:
> 1,2,3,3,5,6,4,4,9,6,11(...). I conjecture that it is essentially
> A003558.
>
> Regards,
>
>
>
> ----- Original Message -----
> From: Vladimir Shevelev <shevelev at bgu.ac.il>
> Date: Friday, July 16, 2010 22:28
> Subject: [seqfan] Re: Pseudo-arithmetic progressions
> To: seqfan at list.seqfan.eu
>
> > As a continuation, I have just submitted also the following
> two
> > sequences:
> > %I A179480
> > %S A179480 1,1,2,1,3,3,2,1,5,2,6,5,5,7,2,1,6,9,6,3,3,6,12
> > %N A179480 A dual sequence to A179382
> > %C A179480 Let m>k>0 be odd numbers. Denote m<->k=A000265(m-
> > k). Then the sequence m<->k, m<->(m<->k), m<-
> >(m<-
> > >(m<->k)),... is periodic. In this sequence, a(n) is the
> > smallest period in case of m=2*n-1,k=1.
> > %e A179480 If n=14, then m=27 and we have 27<->1=13, 27<-
> > >13=7, 27<->7=5, 27<->5=11, 27<->11=1. Thus a(14)=5.
> > %Y A179480 A179382, A179383, A000265
> > %K A179480 nonn
> > %O A179480 2,3
> >
> > %I A179481
> > %S A179481 3,7,11,19,23,29,37,47
> > %N A179481 a(n) = 2*t(n)-1 where t(n) is the sequence of
> records
> > positions of A179480.
> > %C A179481 Question. Whether every term of this sequence is
> > prime?
> > %Y A179481 A179480 A179460 A179382, A179383
> > %K A179481 nonn
> > %O A179481 2,1
> >
> > I call A179480 a dual to A179382, since if to replace all <-
> >
> > by <+> , then, in view of the commutativity of binary
> > operation <+>, we obtain the corresponding pseudo-
> arithmetic
> > progression.
> > Now, of course, to every (0,1)-sequence, one can correspond a
> > sequence of this type by the rule that <+> corresponds to 1
> > and <-> corresponds to 0. It is easy to see that, if a
> given
> > (0,1)-sequence is (eventually) periodic, then the
> corresponding
> > sequence of the considered type will be (eventually) periodic
> as well.
> >
> > Regards,
> >
> >
> > _______________________________________________
> >
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> >
>