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

1,1

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?

Links

Table of n, a(n) for n=1..10.

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;

Prog

(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

Crossrefs

Cf. A000040, A328596, A348268.

Sequence in context: A346686 A309249 A294727 * A007311 A031345 A206702

Adjacent sequences: A348349 A348350 A348351 * A348353 A348354 A348355

Keyword

nonn,more

Author

Thomas Scheuerle, Oct 14 2021

Status

approved