# A006906 question

David Wilson davidwwilson at comcast.net
Mon Aug 20 07:31:05 CEST 2007

```I sent in a comment on A006906 to the effect that it empircally seemed like
lim a(n+1)/a(n) = 3^(1/3). This comment should be ignored or retracted by
NJAS.

The comment should read  lim a(n+3)/a(n) = 3  and something to the effect
that  lim a(n+1)/a(n)  is close to, but not exactly, 3^(1/3).

BTW, thanks, Dean, for the effort.

----- Original Message -----
From: "Dean Hickerson" <dean at math.ucdavis.edu>
To: <seqfan at ext.jussieu.fr>
Sent: Sunday, August 19, 2007 8:36 PM
Subject: Re: A006906 question

> Paul D. Hanna wrote:
>
>> Is it not accurate to state:
>>
>> (*) the limit of a(n+3)/a(n) exists and is equal to 3.
>
> Yes, that follows from my equation (3), since  n mod 3 = (n+3) mod 3.
>
> Dean Hickerson
> dean at math.ucdavis.edu
>
>
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2 related somewhat questions:

Is the sequence of positive integers infinite, where m is included in the
the mth integer from among those positive integers which are coprime to
(m+1)
=
the (m+1)th integer from among those positive integers which are coprime
to m

?

This sequence begins 3, 15,...

For example:
The 3rd positive integer coprime to 4 is 5. (4 is coprime to
1,3,5,7,..The 3rd of these is 5.)
The 4th positive integer coprime to 3 is also 5. (3 is coprime to
1,2,4,5,7,8,... The 4th of these is 5.)

The 15th positive integer coprime to 16 is 29.
The 16th positive integer coprime to 15 is also 29.

Is this sequence of m's (or this sequence with each term added to 1) in
the EIS?

---

I just submitted these two sequences (one of which is an array):

>%S A132422
>1,1,1,1,2,1,1,3,3,1,1,2,4,4,1,1,3,5,7,5,1,1,2,3,4,6,6,1,1,3,4,7,13,17,7,1,
>1,2,5,4,5,6,8,8,1
>%N A132422 Array read by anti-diagonals: a(m,1) = 1, for all positive
>integers m. a(m,n) = the mth integer from among those positive integers
>which are coprime to a(m,n-1).
>%C A132422 Many, if not all, sequences {a(m,n)}, for fixed m, are periodic
>after some point. a(m,n) = a(m,n+A132423(n)) for all n > some integer.
>%e A132422 a(m,n):
>a(m,1):1,1,1,1,1,...
>a(m,2):1,2,3,4,5,...
>a(m,3):1,3,4,7,6,...
>a(m,4):1,2,5,4,13,...
>a(m,5):1,3,3,7,5,...
>For example, the positive integers which are coprime to a(4,2)=4 are
>1,3,5,7,9,... The 4th of these integers is 7. So a(4,3) = 7.
>%Y A132422 A132423
>%O A132422 1
>%K A132422 ,more,nonn,tabl

>%S A132423 1,2,3,2,3,2,3,4,2,2
>%N A132423 a(m) is the smallest positive integer such that A132422(m,n) =
>A132422(m,n+a(m)) for all n > some value.
>%C A132423 Does every sequence A132422(m,n), for fixed m, repeat after
>some point? If there is a prime > m in A132422(m,n) (for fixed m), then
>this sequence repeats.
>%e A132423 A132422(8,n) is 1,8,15,14,17,8,15,14,17,8,15,14,17,... So a(8)
>= 4.
>%Y A132423 A132422
>%O A132423 1
>%K A132423 ,more,nonn,

Is it true that every term of A132423 is finite?

In order for the sequence (for fixed m) A132422(m,n) to never repeat,
there must be no primes > m which occur in the sequence.
This seems like a high hurdle to overcome.

Thanks,
Leroy Quet

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