[seqfan] Re: Pseudo-arithmetic progressions

[Vladimir Shevelev]

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,
Vladimir



----- 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,
> Vladimir
> 
>  Shevelev Vladimir‎
> 
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 Shevelev Vladimir‎