A348352 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.

2, 3, 5, 7, 13, 233, 433, 27361, 121553, 30536929

Offset: 1

Author

Thomas Scheuerle, Oct 14 2021

Keywords

• nonn
• more

Comments

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?

Crossrefs

Cf. A000040, A328596, A348268.

Programs

MATLAB

function a = A348352(max_range)
    a = [];
    bits = floor(log2(max_range))+2;
    p = primes(max_range);
    lw = lyndonwords(1);
    lyndonw = lw{2};
    for n = 2:bits
        lyndonw =[lyndonw lyndonwords(n)];
    end
    for n = 1:length(p)
        prime = p(n);
        wraw = bitget(prime-1,1:bits);
        word = wraw(1:find(wraw == 1, 1, 'last' ));
        if length(lyndonw{n}) == length(word)
            if lyndonw{n} == word
                a = [a prime];
            end
        end
    end
end
function words = lyndonwords(maxlen)
    words = cell(1);
    wordindex = 1;
    w = 0;
    while ~isempty(w)
       len = length(w);
       if(len == maxlen)
            s = [];
            for j = 1:length(w)
                s = [s w(j)];
            end
            words{wordindex} = s;
            wordindex = wordindex + 1;
        else
            while length(w) < maxlen
                 w = [w w(1+length(w)-len)];
            end
       end
        while ~isempty(w) && w(end) == 1
            w = w(1:end-1);
        end
        if ~isempty(w)
            w(end) = 1;
        end
    end
end

Formula

A348268(a(n) - 1) = a(n).

A348268(a(n)*2^m - 1) = a(n)*2^m.

If A000040(m) = a(n) then A328596(m) = a(n) - 1;