A348352 -id:A348352 - cp's OEIS frontend

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.

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

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.

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