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Integer Complexity: Breaking the θ(n2) barrier
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An algorithm with Θ(n log 2 3) as its running time is presented and a proof of the theorem is presented: the largest solutions of f (m) = 3k, 3k±1 are, respectively, m = 3 k , 3 k ± 3 k−1.Abstract:
—The integer complexity of a positive integer n, denoted f (n), is defined as the least number of 1's required to represent n, using only 1's, the addition and multiplication operators, and the parentheses. The running time of the algorithm currently used to compute f (n) is Θ(n 2). In this paper we present an algorithm with Θ(n log 2 3) as its running time. We also present a proof of the theorem: the largest solutions of f (m) = 3k, 3k±1 are, respectively, m = 3 k , 3 k ± 3 k−1 .read more
Citations
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Journal ArticleDOI
Numbers with Integer Complexity Close to the Lower Bound
Harry Altman,Joshua Zelinsky +1 more
TL;DR: In this paper, it was shown that f(2m3k) = 2m + 3k for m ≤ 31 with m and k not both zero, and the same for larger m.
Posted Content
Algorithms for determining integer complexity
J. Arias de Reyna,J. van de Lune +1 more
TL;DR: Three algorithms to compute the complexity of all natural numbers, each superior to the one in [11], are presented and it is shown that they run in time $O(N^\alpha)$ and space $N\log\log N)$.
Dissertation
Integer Complexity, Addition Chains, and Well-Ordering.
TL;DR: Two notions of the “complexity” of a natural number are considered, the first being addition chain length, and the second known simply as “integer complexity”, which is the smallest number of 1’s needed to write n using an arbitrary combination of addition and multiplication.
Journal Article
Integer Complexity: Algorithms and Computational Results.
TL;DR: In this paper, an algorithm for computing stable numbers is presented, which is based on considering the defect of a number, defined by δ(n):=\|n\|-3\log_3 n, building on the methods presented in [3] and [4].
Journal ArticleDOI
A short note on integer complexity
TL;DR: This short note gives a new, constructive upper bound on the smallest number of 1's needed in conjunction with arbitrarily many +, *, and parentheses to write an integer n for generic integers n.
References
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Journal Article
The On-Line Encyclopedia of Integer Sequences.
TL;DR: The On-Line Encyclopedia of Integer Sequences (OEIS) as mentioned in this paper is a database of 13,000 number sequences and is freely available on the Web (http://www.att.com/~njas/sequences/) and is widely used.
Book ChapterDOI
The On-Line Encyclopedia of Integer Sequences
TL;DR: The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used.
Book
Unsolved Problems in Number Theory
TL;DR: In this article, a discussion of hundreds of open questions, organized into 185 different topics, represent aspects of number theory and are organized into six categories: prime numbers, divisibility, additive number theory, Diophantine equations, sequences of integers, and miscellaneous.
Journal ArticleDOI
Unsolved problems in number theory
TL;DR: The topics covered are: additive representation functions, the Erdős-Fuchs theorem, multiplicative problems (involving general sequences), additive and multiplicative Sidon sets, hybrid problems (i.e., problems involving both special and general sequences, arithmetic functions and the greatest prime factor func- tion and mixed problems.
Journal ArticleDOI
Some Suspiciously Simple Sequences
TL;DR: Some Suspiciously Simple Sequences The American Mathematical Monthly: Vol 93, No 3, pp 186-190 as discussed by the authors, was published in 1986, and is a classic example.