Define knk to be the complexity of n, the smallest number of ones needed to write n using an arbitrary combination of addition and multiplication. John Selfridge showed that knk 3 log 3 n for all n. Define the defect of n, denoted by ı.n/, to be knk 3 log 3 n; in this paper we present a method for classifying all n with ı.n/ < r for a given r. From this, we derive several consequences. We prove that k2 m 3 k k D 2m C 3k for m Ä 21 with m and k not both zero, and present a method that can, with more computation, potentially prove the same for larger m. Furthermore, defining A r .x/ to be the number of n with ı.n/ < r and n Ä x, we prove that A r .x/ D ‚ r ..log x/ brcC1 /, allowing us to conclude that the values of knk 3 log 3 n can be arbitrarily large.
Let n be a primitive non-deficient number, withwhere the p i are distinct primes. Let R = p 1 p 2 • • • p k . We prove that there must be an i such that p a i +1 i < 4R 2 . We conjecture that there is always an i such that p a i +1 i < kR and prove this stronger inequality in some cases.
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