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

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, 38, 40, 42, 43, 44, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 61, 62, 64, 65, 66, 68, 70, 72, 73, 74, 76, 77, 78, 80, 82, 84, 86, 87, 88, 90, 92, 93, 94, 95, 96, 98, 100, 102, 103, 104

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. If we take the resulting sequence of binary digits as a fractional real, it can be mapped back into a sequence, as in A092855.

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

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, A093512, A093513, A093514, A093515, A093516, A093517.

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 */}