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

2, 3, 4, 9, 11, 15, 17, 21, 23, 25, 29, 33, 37, 39, 41, 45, 47, 49, 53, 55, 59, 63, 67, 69, 71, 75, 79, 81, 83, 85, 89, 91, 97, 99, 101, 105, 107, 111, 113, 115, 127, 129, 131, 133, 137, 141, 149, 153, 157, 159, 163, 165, 167, 169, 173, 175, 179, 183, 191, 195, 197, 201

Offset: 1

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu)

Keywords

• easy
• nonn

Comments

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.

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

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.

Links

• Ferenc Adorjan, Binary mapping of monotonic sequences - the Aronson and the CA functions
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

Crossrefs

Cf. A092855, A051006, A093510, A093511, A093512, A093513, A093515, A093516, A093517.

Characteristic function for this sequence is A010051(n-2) + A010051(n) (modulo 2). Naturally none of the terms of A006512 occur here.

Programs

PARI

{ca_tr(ca,v)= /* Calculates the Cellular Automaton transform of the vector v by the rule ca */
local(cav=vector(8),a,r=[],i,j,k,l,po,p=vector(3));
a=binary(min(255,ca));k=matsize(a)[2];forstep(i=k,1,- 1,cav[k-i+1]=a[i]);
j=0;l=matsize(v)[2];k=v[l];po=1;
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)));
return(r) /* See the function "isin" at A092875 */}