%I #21 Dec 11 2021 02:11:40
%S 2,3,5,7,13,233,433,27361,121553,30536929
%N Primes p where p-1 is in A328596 (reversed binary expansion is an aperiodic necklace) and the same count of numbers smaller than p-1 are found in A328596 as primes smaller than p exist.
%C If this sequence is infinite, then the density of aperiodic necklaces (Lyndon words) in the reversed binary expansion of numbers and the density of prime numbers, may have some interesting connection. If there exists a deeper relation, an analogy of Goldbach's conjecture based on numbers in A328596 could be investigated, would that provide any new knowledge regarding prime numbers?
%F A348268(a(n) - 1) = a(n).
%F A348268(a(n)*2^m - 1) = a(n)*2^m.
%F If A000040(m) = a(n) then A328596(m) = a(n) - 1;
%o (MATLAB)
%o function a = A348352(max_range)
%o a = [];
%o bits = floor(log2(max_range))+2;
%o p = primes(max_range);
%o lw = lyndonwords(1);
%o lyndonw = lw{2};
%o for n = 2:bits
%o lyndonw =[lyndonw lyndonwords(n)];
%o end
%o for n = 1:length(p)
%o prime = p(n);
%o wraw = bitget(prime-1,1:bits);
%o word = wraw(1:find(wraw == 1, 1, 'last' ));
%o if length(lyndonw{n}) == length(word)
%o if lyndonw{n} == word
%o a = [a prime];
%o end
%o end
%o end
%o end
%o function words = lyndonwords(maxlen)
%o words = cell(1);
%o wordindex = 1;
%o w = 0;
%o while ~isempty(w)
%o len = length(w);
%o if(len == maxlen)
%o s = [];
%o for j = 1:length(w)
%o s = [s w(j)];
%o end
%o words{wordindex} = s;
%o wordindex = wordindex + 1;
%o else
%o while length(w) < maxlen
%o w = [w w(1+length(w)-len)];
%o end
%o end
%o while ~isempty(w) && w(end) == 1
%o w = w(1:end-1);
%o end
%o if ~isempty(w)
%o w(end) = 1;
%o end
%o end
%o end
%Y Cf. A000040, A328596, A348268.
%K nonn,more
%O 1,1
%A _Thomas Scheuerle_, Oct 14 2021