cp's OEIS Frontend

A093511 Transform of the prime sequence by the Rule45 cellular automaton.

%I A093511 #8 Feb 16 2025 08:32:53
%S A093511 1,3,5,6,7,8,10,12,13,14,16,18,19,20,22,24,26,27,28,30,31,32,34,35,36,
%T A093511 38,40,42,43,44,46,48,50,51,52,54,56,57,58,60,61,62,64,65,66,68,70,72,
%U A093511 73,74,76,77,78,80,82,84,86,87,88,90,92,93,94,95,96,98,100,102,103,104
%N A093511 Transform of the prime sequence by the Rule45 cellular automaton.
%C A093511 As described in A051006, a monotonic sequence can be mapped into a fractional real. Then the binary digits of that real can be treated (transformed) by an elementary cellular automaton. If we take the resulting sequence of binary digits as a fractional real, it can be mapped back into a sequence, as in A092855.
%C A093511 Conjecture: For n > 3, the a(n) correspond to the following construct (in numerical order). a(n) terms include "bookend" values at every prime p + 1 (6,8,12,14,18,20,...). Additionally, the values between the bookends are included, unless adjacent to non-"twin composite" bookends. For example, consider bookends 6 and 8. There is only a single value 7 between these, so it is included. This means terms 6, 7 and 8 are included.  Consider bookends 89 + 1, 97 + 1. Ignoring 91 and 97 adjacencies, values 92 through 96 are included. This means terms 90, 92-96 and 98 are included. - _Bill McEachen_, Jun 12 2024
%H A093511 Ferenc Adorjan, <a href="http://web.axelero.hu/fadorjan/aronsf.pdf">Binary mapping of monotonic sequences - the Aronson and the CA functions</a>
%H A093511 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%o A093511 (PARI) {ca_tr(ca,v)= /* Calculates the Cellular Automaton transform of the vector v by the rule ca */
%o A093511 local(cav=vector(8),a,r=[],i,j,k,l,po,p=vector(3));
%o A093511 a=binary(min(255,ca));k=matsize(a)[2];forstep(i=k,1,- 1,cav[k-i+1]=a[i]);
%o A093511 j=0;l=matsize(v)[2];k=v[l];po=1;
%o A093511 for(i=1,k+2,j*=2;po=isin(i,v,l,po);j=(j+max(0,sign(po)))% 8;if(cav[j+1],r=concat(r,i)));
%o A093511 return(r) /* See the function "isin" at A092875 */}
%Y A093511 Cf. A092855, A051006, A093510, A093512, A093513, A093514, A093515, A093516, A093517.
%K A093511 easy,nonn
%O A093511 1,2
%A A093511 Ferenc Adorjan (fadorjan(AT)freemail.hu)