The question “How many 1’s are needed? ” revisited

2009 We present a rigorous and relatively fast method for the computa-tion of the complexity of a natural number (sequence A005245), and answer some old and new questions related to the question in the title of this note. We also extend the known terms of the related sequence A005520. Introduction. The subject of this note was (more or less indirectly) initiated in 1953 by K. Mahler and J. Popken [1]. We begin with a brief description of part of their work: Given a symbol x, consider the set Vn of all formal sum-products which can be constructed by using only the symbol x and precisely n −