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

[Maximilian Hasler]

On Sat, Mar 1, 2008 at 1:06 PM, Ferenc Adorjan <ferencadorjan at gmail.com> wrote:
> Please, check out also the 2nd comment in A093514 by Antti Karttunen. It
> is quite enlightening...

Indeed it summarizes in a clearly understandable way how these
sequences are defined, and seems to indicate that Wilson's observation
can be proved, but yet it does not provide the explicit proof.

I also notice that you pasted  the same PARI code into many sequences,
but on one hand there is always the cross-reference to the "isin"
function which has to be taken elsewhere anyway, and on the other hand
the explicit command to obtain the concrete sequence is not given.
Maybe it would be a better choice to put the general code only in one
place (to make it easier to correct or improve it if desired), but in
a complete, self-contained version, and to give in the different
sequences only a cross-reference to the general code, and rather add
the precise command which produces the given sequence.

PS: I submitted (a correct version without the Vec[t] typo of) the
code I sent you:

 seqruletrans(v,r)={local(c=1,t); r=vecextract(binary(r+256),"-1..-8");
 t=List(vector(v[#v],i,v[c]<i & c++; if(r[1+t=t%4*2+(v[c]==i)],i)));
 listsort(t,1); v=Vec(t); listkill(t); if( v[1], v, vecextract( Vec(v), "^1"))}

as comment to A093515, so maybe it would be a good idea to refer to
that place for the general code. The other sequences may be produced
with

%o A093510 (PARI) A093510=seqruletrans(vector(100,i,prime(i)),30)/*cf.
A093515 for seqruletrans()*/
%o A093511 (PARI) A093511=seqruletrans(vector(100,i,prime(i)),45)/*cf.
A093515 for seqruletrans()*/
%o A093512 (PARI) A093512=seqruletrans(vector(100,i,prime(i)),73)/*cf.
A093515 for seqruletrans()*/
%o A093513 (PARI) A093513=seqruletrans(vector(100,i,prime(i)),89)/*cf.
A093515 for seqruletrans()*/
%o A093514 (PARI) A093510=seqruletrans(vector(100,i,prime(i)),90)/*cf.
A093515 for seqruletrans()*/
%o A093516 (PARI)
A093516=seqruletrans(vector(100,i,prime(i)),137)/*cf. A093515 for
seqruletrans()*/
%o A093517 (PARI)
A093517=seqruletrans(vector(100,i,prime(i)),225)/*cf. A093515 for
seqruletrans()*/

(NB: This function is quite efficient and gives the result for 10^4
primes in < 0.5 sec, but depending on the rule it gives typically 2 to
3 times more terms as result...)

Regards,
Maximilian