A348268 -id:A348268 - cp's OEIS frontend

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

Thomas Scheuerle, Oct 14 2021

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?

Cf. A000040, A328596, A348268.

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

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;

A348369 Number of ways A328596(n) (the reversed binary expansion is an aperiodic necklace) can be expressed as sum A328596(k) + A328596(m) with 0 < k,m < n. The cases A328596(k) + A328596(m) and A328596(m) + A328596(k) are considered equal.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 3, 2, 3, 4, 3, 4, 6, 3, 5, 5, 5, 5, 7, 5, 5, 9, 4, 6, 5, 8, 7, 9, 9, 7, 8, 10, 9, 9, 13, 6, 8, 8, 9, 15, 7, 10, 8, 14, 10, 12, 10, 11, 13, 13, 14, 14, 15, 16, 13, 14, 15, 15, 18, 14, 18, 16, 16, 22, 10, 9, 12, 12, 10, 24, 10, 16, 9, 21, 14, 20, 12

Offset: 1

Thomas Scheuerle, Oct 15 2021

nonn

Conjecture: The only zero in this sequence is a(1). A348268 maps all terms of A328596 bijective to primes. Let P be this bijection between Lyndon words and primes and P' its inverse. Then for each prime q, there exist primes r and s such that q = P(P'(r) + P'(s)). If we were to define a table T(m,n) which encodes the sum q + 1 = (A000040(m) + A000040(n)), then q = P(P'(A000040(m)) + P'(A000040(n))) would be a permutation of this table; this connects this conjecture to Goldbach's conjecture.

All reversed binary expansions of powers of two are Lyndon words. All reversed binary expansions of numbers of the form 2*(2^m - 1) are Lyndon words, too. 2*(2^m - 1) + 2 is again a power of 2. Every positive integer can be expressed as a sum of powers of 2. From this we can conclude that it is always possible to compose terms of A328596(n) (n > 1), as a sum of terms of A328596. This would require at least 2 or more such terms.

			A328596(5) = A328596(2) + A328596(4) = A328596(3) + A328596(3) -> a(5) = 2.
.
Table A: A348268(A348268^-1(m) + A348268^-1(n))
      2   3   5   7
  -----------------
  2| (3)  4   6   8   prime numbers are marked by ()
  3|  4  (5) (7)(11)
  5|  6  (7)(11)  9
  7|  8 (11)  9 (13)
.
Table B: m + n
      2   3   5   7
  -----------------
  2| (4)  5   7   9   prime numbers + 1 are marked by ()
  3|  5  (6) (8) 10
  5|  7  (8) 10 (12)
  7|  9  10 (12)(14)
.
Table B is a permutation of Table A + 1.

Thomas Scheuerle, a(1)..a(4000) (Both axes are logarithmic and denote 2^x and 2^y. It appears that this sequence is self-similar, with an irrational exponent.)

Cf. A328596, A348268, A348352.

cp's OEIS Frontend

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.

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