We consider representing of natural numbers by expressions using 1's, addition, multiplication and parentheses. n denotes the minimum number of 1's in the expressions representing n. The logarithmic complexity n log is defined as n /log 3 n. The values of n log are located in the segment [3, 4.755], but almost nothing is known with certainty about the structure of this "spectrum" (are the values dense somewhere in the segment etc.). We establish a connection between this problem and another difficult problem: the seemingly "almost random" behaviour of digits in the base 3 representations of the numbers 2 n . We consider also representing of natural numbers by expressions that include subtraction, and the so-called P -algorithms -a family of "deterministic" algorithms for building representations of numbers.

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