[seqfan] Re: Periodic Fibonacci-like sequences without multiples of several primes

[Vladimir Shevelev]

With respect to the question "for a given prime p, whether exist one or more  primes greater than p, such that Fibonacci-like sequences  without multiples of these (together with p) primes is periodic? ", I believe that it should be solved in affirmation.
On this topic see my sequences A224523 and A219328. For example, the Fibonacci-like sequence without multiples of primes 659,997,1597,1987 is periodic with period of length 24 (see comment in A219328). Moreover, up to prime(4000) the longest period is of length 198 (see Peter's b-file in A224523). But what methods one can use for research of this problem?
 
Best regards,
Vladimir
 


----- Original Message -----
From: Vladimir Shevelev <shevelev at bgu.ac.il>
Date: Monday, April 1, 2013 6:11
Subject: [seqfan] Periodic Fibonacci-like sequences without multiples of several primes
To: seqfan at list.seqfan.eu

> Dear SeqFans,
>  
> For a given prime p, consider Fbonacci-like numbers {F_p(n)} 
> without moltiples of p which defined in the following way:
>  a(0)=0, a(1)=1, for n>=2, a(n)=a(n-1)+a(n-2), if a(n-
> 1)+a(n-2) is not multiple of p, otherwise, a(n)=a(n-1)+a(n-2) 
> divided by maximal possible power of p. 
>  Note that {F_2(n)} has period of length 1, {F_3(n)} has 
> period of length 3, {F_5(n)} has period of length 6. The first 
> case when such a sequence has not a trivial period and, 
> probably, is non-periodic, is p=7 (A224196). I do not know, if 
> this sequence is bounded? 
> Other questions arise when we consider Fibonacci-like sequences 
> without multiples of several primes which defined quite 
> analogously: e.g., for {F_(p,q)(n)}, a(0)=0, a(1)=1, for n>=2, 
> a(n)=a(n-1)+a(n-2), if a(n-1)+a(n-2) is multiple of neither p no 
> q, otherwise, a(n)=a(n-1)+a(n-2) divided by maximal possible 
> power of p and maximal possible power of q. The first question 
> is the following: for a given prime p, whether exist one or more 
> primes greater than p, such that Fibonacci-like sequences 
> without multiples of these (together with p) primes is periodic? 
> So in case p=7 we could add primes 11 and 13 and obtain an 
> eventually periodic sequence {F_(7,11,13)(n)} with period of 
> length 12: 0,1,1,2,3,5,8,1,9,10,19,29,48,1,1,2,3,5,...
> Note that sequence {F_(11,13,19)(n)} has period of length 9, 
> sequence {F_(13,19,23)(n)} has period of length 12, sequence 
> {F_(17,19,23,29)(n)} has period of length 15, sequence 
> {F_(19,23,31,53,59,89)(n)} has period of length 24,
> while sequence {F_(23,29,73,233)(n)} has period of length 18, etc.
> By another strange observation, all lengths of periods are 
> multiple of 3 (except for trivial case of {F_2(n)} with period 
> 1). Could anyone to find a periodic Fibonacci-like sequences 
> without multiples of several primes with a length of period not 
> multiple of 3?
>  
> Best regards,
> Vladimir
> 
>  Shevelev Vladimir‎
> 
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 Shevelev Vladimir‎