1,1

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.

The "Rule110" transform as used here involves a right-shift of the sequence before applying the transform as described at http://mathworld.wolfram.com/Rule110.html. - M. F. Hasler (MHasler(AT)univ-ag.fr), Mar 01 2008

Robert G. Wilson V observed that this sequence contains exactly the indices for which A121561 equals 1. - M. F. Hasler (MHasler(AT)univ-ag.fr), Mar 01 2008

M. F. Hasler, Table of n, a(n) for n=1,...,19998 (using prime(1..10^4)).

Ferenc Adorjan, Binary mapping of monotonic sequences - the Aronson and the CA functions

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

Eric Weisstein's World of Mathematics, Rule110 Elementary Cellular Automaton

(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 */}

(PARI) A093515=seqruletrans(vector(10^4, i, prime(i)), 110) - M. F. Hasler (MHasler(AT)univ-ag.fr), Mar 01 2008

(PARI) /* transform a sequence v by the rule r - Note: v could be replaced by a function, e.g. v[c] => prime(c) here */ seqruletrans(v, r)={local(c=1, t); r=vecextract(binary(r+256), "-1..-8"); t=List(vector(v[ #v], i, v[c]<i & c++; if(r[1+t=t%4*2+(v[c]==i)], i))); listsort(t, 1); v=Vec(t); listkill(t); if( v[1], v, vecextract( Vec(v), "^1"))} - M. F. Hasler (MHasler(AT)univ-ag.fr), Mar 01 2008

(PARI) A121561_is_1(n)=vector(n+n=0, i, while(!isprime(n++)&!isprime(n-1), ); n) - M. F. Hasler (MHasler(AT)univ-ag.fr), Mar 01 2008

Cf. A092855, A051006, A093510, A093511, A093512, A093513, A093514, A093516, A093517.

Cf. A121561.

Sequence in context: A032847 A023778 A053577 this_sequence A084369 A167211 A067133

Adjacent sequences: A093512 A093513 A093514 this_sequence A093516 A093517 A093518

easy,nonn

Ferenc Adorjan (fadorjan(AT)freemail.hu)