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%I #28 Apr 01 2019 08:33:12
%S 1,2,3,4,5,5,7,8,9,10,11,9,11,11,15,16,17,18,19,18,21,21,23,17,19,21,
%T 23,19,23,23,31,32,33,34,35,36,37,37,39,34,37,42,43,37,43,43,47,33,35,
%U 37,39,37,43,43,47,35,39,43,47,39,47,47,63,64,65,66,67,68,69,69,71,68,73
%N Smallest number "c-equivalent" to n.
%C 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.
%C 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.
%C 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.
%C 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.
%e The set of integers c-equivalent to 38 is {37,38,41,44,50,52} (with the binary representations 100101, 100110, 101001, 101100, 110010, and 110100, respectively). The smallest of these numbers is 37. Thus, a(38) = 37. Alternatively, the substrings of 100110_binary = 38 correspond to writing 6 as the sum of 3+1+2, which is a permutation of the partition 6 = 3+2+1, where the right hand side corresponds to 37. (On the other hand, only 41 and 52 may be achieved from 38 by cyclic permutations of the bits, whence A163382(38) = 38.)
%Y A114994 = range(a), A233312(n) = a(A020330(n)).
%Y Cf. also A003714, A118113, A163382, A233249.
%K nonn,easy
%O 1,2
%A _Jörgen Backelin_, Jan 16 2016
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