cp's OEIS Frontend

A093513 Transform of the prime sequence by the Rule89 cellular automaton.

%I A093513 #7 Feb 16 2025 08:32:53
%S A093513 1,3,4,9,10,15,16,21,22,25,26,27,28,33,34,35,36,39,40,45,46,49,50,51,
%T A093513 52,55,56,57,58,63,64,65,66,69,70,75,76,77,78,81,82,85,86,87,88,91,92,
%U A093513 93,94,95,96,99,100,105,106,111,112,115,116,117,118,119,120,121,122,123
%N A093513 Transform of the prime sequence by the Rule89 cellular automaton.
%C A093513 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 A093513 For n > 3, the a(n) are those missed by the following construct. Start with b(0) = 1. If b(n-1) is not prime, b(n) is the next prime, otherwise b(n) is the next integer. This yields 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 17, 18, 19, 20, 23, ... with missed values a(n) = 9, 10, 15, 16, 21, 22, .... Interestingly, b(n) appears to be A093515 for n>0. This fits with Karttunen's observation at A093515. - _Bill McEachen_, Jun 12 2024
%H A093513 Ferenc Adorjan, <a href="http://web.axelero.hu/fadorjan/aronsf.pdf">Binary mapping of monotonic sequences - the Aronson and the CA functions</a>
%H A093513 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%o A093513 (PARI) {ca_tr(ca,v)= /* Calculates the Cellular Automaton transform of the vector v by the rule ca */
%o A093513 local(cav=vector(8),a,r=[],i,j,k,l,po,p=vector(3));
%o A093513 a=binary(min(255,ca));k=matsize(a)[2];forstep(i=k,1,- 1,cav[k-i+1]=a[i]);
%o A093513 j=0;l=matsize(v)[2];k=v[l];po=1;
%o A093513 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 A093513 return(r) /* See the function "isin" at A092875 */}
%Y A093513 Cf. A092855, A051006, A093510, A093511, A093512, A093514, A093515, A093516, A093517.
%K A093513 easy,nonn
%O A093513 1,2
%A A093513 Ferenc Adorjan (fadorjan(AT)freemail.hu)