RE First differences are primes

[Max A.]

On 9/25/06, Tautócrona <tautocrona at terra.es> wrote:
> >could someone please extend this (if of interest for the OEIS) :
> >S = 1 4 6 25 30 77 84 95 108 125 148 177 208 245 286 329 382 441 502 573 640 713 792 875
> >964 1065 1162 ...
> >Definition :
> >« Non-primes sequence whose first differences show all primes, once »
>
> I think the related seq: "primes that appear in seq above, in its order" is also of
> interest. From your own example, the first terms would be:
>
> 3, 2, 19, 5...

First 50 differences:

3, 2, 19, 5, 47, 7, 11, 13, 17, 23, 29, 31, 37, 41, 43, 53, 59, 61,
71, 67, 73, 79, 83, 89, 101, 97, 103, 107, 113, 109, 131, 127, 137,
139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211,
223, 229, 227

> Would every prime appear on it? Which is the most expected deviation from the original
> place for that prime? This is, if a(n) is the n-th term of this seq,
> is |a(n)-Pi(a(n)| bounded? How does a(n)/Pi(a(n)) grow?

The n-th prime appear in the sequence of first differences not later
than at 2n-th position.
To prove that it is enough to notice that in the original sequence
(excluding the first element) odd and even numbers alternate.
Therefore, from each odd elements m the sequence simply jumps to an
even element m+p where p is the smallest previously unused prime.

Max