[seqfan] Re: A sequence closely related to A140525

[Maximilian Hasler]

On Mon, Feb 9, 2009 at 8:23 AM, Peter Pein <petsie at dordos.net> wrote:
> a(1) = 3,
> a(n + 1) = 1 + a(n) + least odd primefactor of a(n)
>
> which is apparently A140525 + 1.

indeed :

a(n) = 1+b(n) with b(n)=2,

b(n+1)=1+b(n)+ least odd prime factor of (b(n)+1)

= 1+b(n)+ least prime factor of (b(n)+1)
   [assuming that as b(n) is even, which is then
   again the case for b(n)+1 and thus forever]

= least integer > b(n)+1 not coprime to b(n)+1
  [since such an integer must share at least
  one prime factor p with b(n)+1 and thus
  be of the form b(n)+1 + kp]

and since it is even it is also not coprime to b(n)

thus (b(n)) satisfies the definition of A140525.

(The ">=..." in the definition of A140525 can obviously be changed to
"> 1+..."  since the successor of a b(n) is coprime to b(n) and the
successor of b(n)+1 is coprime to b(n)+1.)

Maximilian