%I #45 May 06 2022 13:13:51

%S 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,

%T 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,

%U 18,14,18,16,16,22,10,9,12,12,10,24,10,16,9,21,14,20,12

%N 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.

%C 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.

%C 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.

%H Thomas Scheuerle, <a href="/A348369/a348369.svg">a(1)..a(4000)</a> (Both axes are logarithmic and denote 2^x and 2^y. It appears that this sequence is self-similar, with an irrational exponent.)

%e A328596(5) = A328596(2) + A328596(4) = A328596(3) + A328596(3) -> a(5) = 2.

%e .

%e Table A: A348268(A348268^-1(m) + A348268^-1(n))

%e 2 3 5 7

%e -----------------

%e 2| (3) 4 6 8 prime numbers are marked by ()

%e 3| 4 (5) (7)(11)

%e 5| 6 (7)(11) 9

%e 7| 8 (11) 9 (13)

%e .

%e Table B: m + n

%e 2 3 5 7

%e -----------------

%e 2| (4) 5 7 9 prime numbers + 1 are marked by ()

%e 3| 5 (6) (8) 10

%e 5| 7 (8) 10 (12)

%e 7| 9 10 (12)(14)

%e .

%e Table B is a permutation of Table A + 1.

%Y Cf. A328596, A348268, A348352.

%K nonn

%O 1,5

%A _Thomas Scheuerle_, Oct 15 2021