[seqfan] A254077
john.mason at lispa.it
john.mason at lispa.it
Mon May 30 14:28:01 CEST 2016
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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