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For c-equivalence, see the comments to A233249. Briefly put, two positive integers m and n are c-equivalent in the sense of Vladimir Shevelev, if they have ordinary binary representations with the same multisets of substrings resulting from cutting the full strings immediately before each bit 1. A(n) is defined as the smallest positive integer, which is c-equivalent to n. Alternatively, in the manners of A114994, the lengths of these substrings can be considered as representing ways to write integers as sums of positive integers with arbitrarily ordered sums, and a(n) as the unique integer whose corresponding substring lengths form the corresponding integer partition.

For instance, the ordinary binary representations of 11, 13, and 14 are 1011, 1101, and 1110, respectively, which yields the equal multisets {"10","1","1"}, and {"1","10","1"}, and {"1","1","10"} of strings, respectively; whence 11, 13, and 14 are c-equivalent.

A(n) is an odd number if and only if the substring "1" appears at least once in the multiset. Since this is the case if and only if it also holds for the binary representation of n concatenated with itself, and A233312(n) = a(m) for the number m whose binary representation is this concatenation, we have a(n) == A233312(n) (mod 2) for all n. Moreover, empirical data has suggested that perhaps A233312(n)+1 == A171791(n+1) (mod 2) for all n >= 1. This relation holds in general if and only if a(n)+1 == A171791(n+1) for the same n, which in its turn is true if and only if the relation with fibbinary numbers first empirically observed by Paul D. Hanna in the comments to A171791 holds in general.

The sequence A163382 also maps n to a c-equivalent integer <=n; however, here, only cyclic permutations of the sequences of substrings are allowed. Thus, a more restricted equivalence relation is used; whence a(n) <= A163382(n) for all n. Equality holds for infinitely many n, including n = 1..37.

The smallest positive number a(n), such that any triangle-free cyclic (also known as circulant) graph with a(n) vertices has independence number at least n. The terminology and the terms given here are from Harborth and Krause.

a(n) <= A267295(n) for all n.

Moreover, the sequence is related to the ordinary two-color Ramsey numbers R(3,n), given in A000791, by the relation a(n) <= A000791(n) for all n, as proved by Kalbfleisch. This inequality is known to be an equality for n = 2 or 4 <= n <= 5.

The smallest number a(n), such that any triangle-free cyclic (also known as circulant) graph with at least a(n) vertices has independence number at least n. The terminology and the terms given here are from Harborth and Krause (2003); however, in another form, essentially they were considered and partly calculated already by Kalbfleich in 1965.

a(n) = A000789(n)+1 and a(n) >= A267296(n) for all n.

Moreover, the sequence is related to the ordinary two-color Ramsey numbers R(3,n), given in A000791, by the relation a(n) <= A000791(n) for all n, as proved by Kalbfleisch. This inequality is known to be strict for 6 <= n <= 8, and for n = 10.

For n <= 8, a(n) is the Turan number T(3,n), realized by a complete bipartite graph K(m, m) if n = 2m or K(m, m+1) if n = 2m+1, since then that graph has maximal degree <= 4.

For n >= 10, any 4-regular triangle-free graph (i.e., graph with no three-cliques) will do.

For n = 9, there is no 4-regular triangle-free graph, as can be seen from inspection.