Curious relation: (Where A121561(n) = 1) == A093515

[Max Alekseyev]

On Fri, Feb 29, 2008 at 3:58 PM, Max Alekseyev <maxale at gmail.com> wrote:
> First off, the numbers n such that A121561(n)=1 are exactly numbers of
>  the form n=p or n=p+1, where p is a prime number.
>
>  Correspondingly, the numbers n such that A121561(n)=2 are exactly
>  numbers of the form n=q+p or n=q+p+1, where p and q are prime numbers
>  such that there are no primes in the interval [q+1,n].
>
>  Therefore, if Bob's observation is true, it implies a simpler
>  description for A093513 and A093515.
>  But I have not followed yet the definition of A093513 or A093515 to
>  justify that.

Here is a rather straightforward proof that A093515 is indeed a
sequence of primes and primes increased by 1.

Note that the primes, except initial 2 and 3, are never close to each
other. Therefore, we need to consider only rules 0,1,2,4,5 enumerated
from right to left as shown at
http://mathworld.wolfram.com/Rule110.html (i.e., rules with no two
neighboring black squares).

btw, if we want our input sequence of bits to be consistent with the
representation at MathWorld (see the link above), we should assume
that it is written
from right to left.

Rules 0 and 4 do not set up any new unit bits in the output, Rules 1
and 5 set up an unit bit right after each unit bit (i.e., they bring
p+1 in the output sequence for every prime p in the input sequence),
and Rule 3 simply copies existing bits from the input to the output.
To complete the proof it's enough to notice that for bits 2,3 Rules
2,3 are applied and these bits are simply copied to the output.

I believe the suggested formula for A093513 can be proved similarly
(if it is correct of course).

I also suggest the following rather simple PARI code for computing
sequences of this type:

{ ruletrans(v,r) = local(t=0,u=[]); for(i=1,#v, t=(t%4)*2+v[i]; if(
bittest(r,t), u=concat(u,[i])) ); u }

with sample functions:

{ A093515() = ruletrans(vector(1000,i,isprime(i)),110) }
{ A093513() = ruletrans(vector(1000,i,isprime(i)),89) }

Note that it not exactly a transformation of sequences since it takes
an indicator 0-1 values and outputs the sequence (of the indices of
unit bits), but for computing sequences in OEIS it is more than
enough.

Regards,
Max