[seqfan] Re: Rif: Re: Rif: A light variation

[M. F. Hasler]

On Thu, Jan 21, 2016 at 8:08 AM, <john.mason at lispa.it> wrote:

> Following Maximilian's idea of looking at the late birds, I generated the
> sequence through 10^5 terms.
> Here are the first 18 late birds, that is numbers not present in the
> sequence.
> After each one I show the number of composites. These high numbers, higher
> than the digit sum of 99999, are a hindrance to the late birds being
> picked.
> An exception is 25470 which is followed by a prime, and then 51
> composites. But until the term 100000 itself is included in the sequence,
> we won't have a number with a digit sum of 1.
>

Yes, the same happens already with the numbers 1326 and 6912 I mentioned:
the gap to the next prime (here 5) is smaller than the available digit sums
(up to the next power of ten), and the step to the subsequent prime is very
large (here 35 which can only be 8999 or permutations).
However, there *are* infinitely many numbers with arbitrarily large digital
sums, and we know that our "late birds" keep sitting there until the next
of this kind appears in the sequence, to be its successor.

So (it this a "proof"?), all numbers will eventually appear in this variant
(which is now A267308).

Maximilian


On Wed, Jan 20, 2016 at 4:21 PM, <john.mason at lispa.it> wrote:

>
> A variation on Eric's theme could be (if not already present) :
> a(n) is smallest positive integer not already in a() such that a(n) +
the
> sum of the digits of a(n-1) is prime. a(1)=1.