[seqfan] Re: A098550.

[Bob Selcoe]

Hi Ed and seqfans;

I especially like the first of these sequences; unless I'm mistaken the 
conjecture is true.

Time doesn't permit a formal proof (it's late here in Texas!), but the basic 
idea is that:

i)  The primes Pi(j) = a(n) occur in ascending order, with Pi(j) first 
occurring as a factor in a composite a(n-2).
ii) The composites in the sequence will be comprised of prime factors which 
already occur in the sequence OR include - exactly once - the next prime 
(Pi(j+1)) as a factor.

Therefore all the odd numbers ultimately must occur in the sequence.

I think a formal proof would be useful to include with the sequence.

Cheers,
Bob Selcoe

This means --------------------------------------------------
From: "L. Edson Jeffery" <lejeffery2 at gmail.com>
Sent: Monday, December 01, 2014 10:17 PM
To: <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: A098550.

> Also, two more related sequences could be added to the database, if there
> is any interest:
>
> (1)  Restricting to the odd natural numbers,
>
> A(n) = 2*n-1, if n <= 3;
> A(n) = least odd k not already in A such that GCD(k,A(n-1))=1 and
> GCD(k,A(n-2))>1, otherwise.
>
> A begins
>
> {1, 3, 5, 9, 25, 21, 55, 7, 11, 35, 33, 49, 15, 77, 27, 91, 45, 13, ...},
>
> and is conjectured to be a permutation of the odd natural numbers.
>
> (2)  Then there is the "index" sequence B corresponding to A:
>
> B(n) = (A(n)+1)/2.
>
> B begins
>
> {1, 2, 3, 5, 13, 11, 28, 4, 6, 18, 17, 25, 8, 39, 14, 46, 23, 7, ...},
>
> and is conjectured to be a permutation of the natural numbers.
>
> I would prefer to leave this project for someone with a better grasp of
> A098550.
>
> Ed Jeffery
>
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