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

1, 3, 4, 9, 10, 15, 16, 21, 22, 25, 26, 27, 28, 33, 34, 35, 36, 39, 40, 45, 46, 49, 50, 51, 52, 55, 56, 57, 58, 63, 64, 65, 66, 69, 70, 75, 76, 77, 78, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 99, 100, 105, 106, 111, 112, 115, 116, 117, 118, 119, 120, 121, 122, 123

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.

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

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, 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 */}