cp's OEIS Frontend

A093514 Transform of the prime sequence by the Rule90 cellular automaton.

%I A093514 #7 Feb 16 2025 08:32:53
%S A093514 2,3,4,9,11,15,17,21,23,25,29,33,37,39,41,45,47,49,53,55,59,63,67,69,
%T A093514 71,75,79,81,83,85,89,91,97,99,101,105,107,111,113,115,127,129,131,
%U A093514 133,137,141,149,153,157,159,163,165,167,169,173,175,179,183,191,195,197,201
%N A093514 Transform of the prime sequence by the Rule90 cellular automaton.
%C A093514 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. Taken resulted sequence of binary digits as a fractional real, it can be mapped back into a sequence, as in A092855.
%C A093514 n is in this sequence if either n-2 OR n is prime but not both. Similar simple propositional rules can be given for all "RuleXXX" transforms of primes (or any strictly monotone sequence with a well-defined characteristic function) because the idea in these sequences is to take the characteristic function, consider it as an infinite binary word, apply one generation of some one-dimensional cellular automaton rule "XXX" to it and define the new sequence by this characteristic function. - _Antti Karttunen_, Apr 22 2004
%C A093514 For example, 2 is included because 0 is not prime, but 2 is. 3 is included because 1 is not prime, but 3 is. 4 is included because 2 is prime, although 4 is not. 5 is not included because both 3 and 5 are primes, 9 is included because 7 is prime, but 9 is not.
%H A093514 Ferenc Adorjan, <a href="http://web.axelero.hu/fadorjan/aronsf.pdf">Binary mapping of monotonic sequences - the Aronson and the CA functions</a>
%H A093514 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%o A093514 (PARI) {ca_tr(ca,v)= /* Calculates the Cellular Automaton transform of the vector v by the rule ca */
%o A093514 local(cav=vector(8),a,r=[],i,j,k,l,po,p=vector(3));
%o A093514 a=binary(min(255,ca));k=matsize(a)[2];forstep(i=k,1,- 1,cav[k-i+1]=a[i]);
%o A093514 j=0;l=matsize(v)[2];k=v[l];po=1;
%o A093514 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 A093514 return(r) /* See the function "isin" at A092875 */}
%Y A093514 Cf. A092855, A051006, A093510, A093511, A093512, A093513, A093515, A093516, A093517.
%Y A093514 Characteristic function for this sequence is A010051(n-2) + A010051(n) (modulo 2). Naturally none of the terms of A006512 occur here.
%K A093514 easy,nonn
%O A093514 1,1
%A A093514 Ferenc Adorjan (fadorjan(AT)freemail.hu)