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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.
1
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
1,5
COMMENTS
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.
LINKS
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.)
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn
AUTHOR
Thomas Scheuerle, Oct 15 2021
STATUS
approved