[seqfan] A254077

[john.mason at lispa.it]

Dear Seqfans
I wonder if anyone is interested in proving any of the following 
conjectures about A254077.
It's definition is: a(n) = n if n <= 3, otherwise the smallest number not 
occurring earlier such that gcd(a(n),a(n-2)) > gcd(a(n),a(n-1)) 

Some of the following are present in the OEIS page. Some are not, and they 
are "obviously" true. The problem is in the apostrophes. The last one 
looks the easiest.

1. The sequence is infinite.
2. The sequence contains all natural numbers.
3. Except for n=2,3,21 corresponding to primes 2,3,11, if a(n) is prime, 
then a(n-2) = 2*a(n). 
4. Except for n=2,3,21 corresponding to primes 2,3,11, if a(n)=mp is the 
first occorrence of prime p as a factor, then m=2 and a(n+2)=p.
5. Except for n=2,3 corresponding to primes 2,3, if a(n)=mp is the first 
occorrence of prime p as a factor, then p does not divide a(n+1).

john

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