The Kitchen Sink and Other Oddities

Atabey Kaygun

Landau's Function

The problem I was working on yesterday is an old problem that was first studied by Landau. I recommend an excellent article on the subject written by W. Miller.

If we use \(g(n)\) the denote the maximum order of an element in \(\Sigma_n\) the symmetric group on \(n\)-letters, the sequence is already recorded at the On-Line Encyclopedia of Integer Sequences.

There are different implementations calculating \(g(n)\). I recommend J.-L. Nicolas’ recent paper at arXiv for a more recent overview and references.

There are really fast and efficient implementations out there. There is one implementation that I really liked. Today, I will re-write it in lisp by changing it a bit along the way.

An implementation

For my implementation, I will need a function that returns the first prime number appearing after a specific integer. This is not very efficient but it will do.

(defun next-prime (n)
  (cond 
    ((< n 2) 2)
    ((= n 2) 3)
    ((evenp n) (next-prime (1- n)))
    (t (let* ((m (+ 2 n))
              (z (floor (sqrt m))))
          (do* ((a 3 (+ a 2)))
               ((or (> a z) (zerop (mod m a)))
                (if (> a z) 
                   m 
                   (next-prime m))))))))

NEXT-PRIME

Next, I took the core of the implementation I mentioned and translated as follows:

(defun landau (n)
  (let* ((limit (floor (* 1.328 (sqrt (* n (log n))))))
         (acc (make-array n :initial-element 1)))
      (do ((p 2 (next-prime p)))
          ((> p limit) (aref acc (1- n)))
         (loop for m from (1- n) downto 0 do
            (setf (aref acc m)
                  (do* ((res (aref acc m)
                             (max res (* q (aref acc (- m q)))))
                        (q p (* p q)))
                  ((> q m) res)))))))

LANDAU

And let me test it:

 (loop for i from 1 to 200 collect (landau i))

(1 1 2 3 4 6 6 12 15 20 30 30 60 60 84 105 140 210 210 420 420 420 420 840
 840 1260 1260 1540 2310 2520 4620 4620 5460 5460 9240 9240 13860 13860
 16380 16380 27720 30030 32760 60060 60060 60060 60060 120120 120120 180180
 180180 180180 180180 360360 360360 360360 360360 471240 510510 556920
 1021020 1021020 1141140 1141140 2042040 2042040 3063060 3063060 3423420
 3423420 6126120 6126120 6846840 6846840 6846840 6846840 8953560 9699690
 12252240 19399380 19399380 19399380 19399380 38798760 38798760 58198140
 58198140 58198140 58198140 116396280 116396280 116396280 116396280
 140900760 140900760 157477320 157477320 232792560 232792560 232792560
 232792560 281801520 446185740 446185740 446185740 446185740 892371480
 892371480 1338557220 1338557220 1338557220 1338557220 2677114440
 2677114440 2677114440 2677114440 2677114440 2677114440 3375492120
 3375492120 5354228880 5354228880 5354228880 5354228880 5354228880
 5354228880 6750984240 6750984240 7216569360 7216569360 8172244080
 12939386460 13385572200 13831757940 13831757940 25878772920 25878772920
 38818159380 38818159380 41495273820 41495273820 77636318760 77636318760
 82990547640 82990547640 82990547640 82990547640 82990547640 82990547640
 155272637520 155272637520 165981095280 165981095280 165981095280
 165981095280 165981095280 165981095280 209280511440 209280511440
 232908956280 232908956280 388181593800 401120980260 414952738200
 414952738200 414952738200 802241960520 802241960520 1203362940780
 1203362940780 1203362940780 1203362940780 2406725881560 2406725881560
 2406725881560 2406725881560 2406725881560 2406725881560 2872543794120
 2872543794120 4813451763120 4813451763120 4813451763120 4813451763120
 4813451763120 4813451763120 5745087588240 5745087588240 6141300525360
 6141300525360 7220177644680 7220177644680 12033629407800 12033629407800
 12033629407800 12033629407800 12033629407800 12033629407800 14440355289360
 14841476269620)

Older Posts

[2024-11-12] Hosoya Index of Balanced Binary Trees

[2024-11-11] Hosoya Index of a Graph

[2024-10-29] The Clique Number of a Simple Graph

[2024-10-28] The Size of Maximally Independent Subsets in a Graph

[2023-11-03] Graph Algorithms in JGraphT with Common Lisp

[2023-10-28] An Implementation of Pandas’ cut and qcut in Lisp

[2023-07-24] A Collatz-like Conjecture for the Projective Line

[2023-03-06] Twin Primes, Cousin Primes, Sexy Primes, and Prime Triplets

[2023-03-02] Set of All Partitions of a Finite Set

[2023-02-14] Non-crossing Partitions and Dyck Words

[2023-02-13] Non-crossing Linear Chords

[2023-02-04] Clojure/Python Interop Examples

[2023-01-14] Graph Algorithms in Clojure with JGraphT

[2022-03-29] 2D-Random Walk

[2022-03-28] Trade Deficit vs Exchange Rate Curve

[2022-03-16] Working with World Bank Data in Python

[2022-03-09] Working with European Central Bank data in python (revisited)

[2022-01-24] A Clique Analysis of Quakers in early modern Britain (1500-1700)

[2021-12-05] Boyer–Moore and Misra-Gries Algorithms in Clojure

[2021-09-12] Tension in Text Plotted

[2021-09-02] Statistical Distributions using Apache Commons Math in Clojure

[2021-08-31] Reduce with Intermediate Results in Common Lisp

[2021-08-21] Multivariate Regression Implemented in Clojure

[2021-05-29] Using Neural Networks to Detect Graph Properties

[2021-04-17] Fast Null-Space Calculation via LU-Decomposition

[2021-02-24] Stoer-Wagner Algorithm in Clojure

[2021-02-19] Calculating Vertex Covers in Clojure

[2021-02-18] Listing All Paths in a Graph

[2021-02-14] Strict Dyck Words and Fibonacci Numbers

[2021-02-14] Kruskal’s Algorithm in Common Lisp

[2021-02-13] Kruskal's Algorithm Implemented in Clojure

[2021-02-10] An integer dynamical system of integers

[2021-02-08] Binary Symmetrization

[2021-01-28] Prüfer Encoding and Decoding of a Tree in Clojure

[2021-01-27] Counting Cycle-Free Paths in a Graph

[2021-01-27] Counting Connected Labeled Graphs

[2020-12-18] Counting Graphs with a Prescribed Degree Sequence

[2020-12-13] Havel–Hakimi Algorithm in Clojure

[2020-12-12] Havel–Hakimi Algorithm in Common-Lisp

[2020-10-23] The Quadratic Casimir Element

[2020-07-04] Collatz Sequence in Binary

[2020-07-02] A Lazy Sequence of Primes in Clojure

[2020-06-10] Yet Another Fizz-Buzz in Common Lisp

[2020-05-12] ECB Data with Clojure and Vega-Lite

[2020-05-06] Processing ECB Data with Common Lisp

[2020-04-17] Next Permutation in the Lexicographical Ordering

[2020-04-13] Turkish Hyphenation in Common Lisp

[2020-04-01] Using JavaPlex with Clojure

[2019-11-05] Constricted Arithmetic Progressions

[2019-11-03] The Number of Arithmetic Progressions of Integers

[2019-05-06] Bron-Kerbosch Algorithm in Clojure

[2019-05-01] An Implementation of Ford-Fulkerson Algorithm in Clojure

[2019-04-22] Document Summarization via Nonnegative Matrix Factorization

[2019-04-20] Latent Semantic Analysis in Clojure

[2019-04-13] K-Nearest Neighbors Algorithm in Clojure

[2019-04-06] K-Means Implemented in Clojure

[2019-03-19] Prüfer Encoding/Decoding of a Tree in Common Lisp

[2019-03-05] Gale-Shaply Algorithm in Common Lisp

[2019-03-02] Calculating The Correct Rank of a Matrix

[2018-12-04] Feed-forward and back-propagation in neural networks as left- and right-fold

[2018-10-31] Nonnegative Matrix Decomposition in Clojure

[2018-10-30] Non-negative Matrix Decomposition in Scala

[2018-08-30] Working with European Central Bank Data in Scala

[2018-07-30] Perverse Sequences

[2018-05-28] Online Perceptron in Common Lisp

[2018-05-26] Online Perceptron

[2018-05-18] Online Regression

[2018-05-06] Knut’s Algorithm-S in Common Lisp

[2018-02-28] Irreducible Dyck Words

[2018-02-19] Optimization with GNU Scientific Library for Lisp

[2018-02-10] Van Eck’s Sequence

[2018-02-09] Hiring networks in mathematics

[2018-02-08] Linus Sequence

[2018-02-05] Egyptian Fractions

[2018-02-01] Listing all Young Tableaux

[2018-01-23] Collatz sequence (yet again)

[2018-01-15] Hofstadter's Q sequence

[2018-01-09] Farey Sequence

[2018-01-09] Catalan's Triangle

[2018-01-06] The Shoelace Formula for the Area of a Polygon

[2017-10-01] Working with European Central Bank Data in Python

[2017-09-27] Expected Value of the Diameter of a Tree

[2017-09-26] Using Quandl with kixi.stats on Clojure

[2017-09-22] Using Quandl with Common Lisp

[2017-08-05] Solving Linear Equations in Natural Numbers

[2017-07-31] Transitive Closure of a Directed Graph or a Relation

[2017-07-20] Steenrod-Milnor and Tournament Sequences

[2017-07-15] A lower bound on the radius of a graph

[2017-07-08] All partitions of an integer

[2017-07-06] Some Hasse Diagrams

[2017-07-04] Shuffles

[2017-07-03] Kaprekar Sequence

[2017-07-01] Lattice of Dyck Words

[2017-06-28] The poset of connected subgraphs of a connected graph

[2017-06-21] Calculating the Diameter and the Radius of a Graph Using Tropic Linear Algebra

[2017-06-19] Generating random regular graphs

[2017-06-14] Estimating the maximum element of a large list

[2017-06-09] A Stochastic Gradient Descent Implementation in Clojure

[2017-06-06] A topology problem

[2017-04-22] Listing duplicate files

[2017-03-14] My First Idris Proof

[2016-12-02] Distinguishing hash functions (part II)

[2016-12-01] Distinguishing hash functions

[2016-10-20] Hofstadter-Conway $10,000 sequence

[2016-08-18] A Solution for Problem 171 of 4Clojure

[2016-08-16] Puzzles and Group Theory

[2016-08-13] Using Weka within Lisp

[2016-07-12] Funniest and Unfunniest Jokes in the Jester Dataset

[2016-07-05] Generating Uniformly Random Connected Graphs

[2016-06-16] The Robinson-Schensted Algorithm

[2016-06-01] Conjugate Partitions

[2016-04-27] Using Word2Vec from Clojure

[2016-04-24] Using Word2Vec from Common Lisp

[2016-04-18] A Migration Analysis

[2016-04-11] Basic Data Analysis with CL without Frameworks

[2016-03-25] Parallel map-reduce in Common Lisp

[2016-02-22] Text Summarization and Topic Analysis

[2016-01-27] Set Covering Problem

[2016-01-25] Kolmogorov-Smirnov Test

[2016-01-20] Eigen-values of the Laplacian and Connected Components of a Graph

[2015-12-12] Dual Graphs

[2015-10-26] Longest Increasing Subsequence Revisited

[2015-10-16] Document Summarization via Markov Chains

[2015-10-07] Computational Literary Analysis

[2015-09-30] Library of Babel in Common Lisp

[2015-09-28] Merging Association Lists in Common Lisp

[2015-07-22] Cheapest Paths via Tropic Matrices

[2015-07-21] Hidden Markov Models via Tropic Matrices

[2015-07-08] A non-technical post

[2015-06-28] An implementation of the Viterbi algorithm in Common Lisp

[2015-05-28] Greatest Common Divisor of Two Rational Numbers

[2015-05-21] Partitions of Equal Measure Whatever the Measure May Be

[2015-05-14] Finding Cliques in a Graph

[2015-05-12] Set Cover Problem

[2015-05-03] Threading Macros in Common Lisp

[2015-05-03] Happy Numbers

[2015-05-01] Collatz Primes

[2015-04-23] Splitting Streams

[2015-04-06] Hamming Distance and Double Hashing

[2015-04-05] Hamming Distance and Hashing Functions

[2015-04-05] Hamming Derivative of Hashing Functions

[2015-04-02] A Topology Problem

[2015-03-21] Curve Fitting is a Gram-Schmidt Reduction

[2015-03-08] Maximum number of characters using keystrokes A, Ctrl+A, Ctrl+C and Ctrl+V

[2015-03-06] Eccentricity, Radius and Diameter in a Graph, Revisited

[2015-03-01] Graphs and Entropy

[2015-02-22] Math PhD Hiring Network (Part 3)

[2015-02-19] Math PhD Hiring Network (Part 2)

[2015-02-18] Math PhD Hiring Network (Part 1)

[2015-02-17] Faculty Networks and Inequality in Hiring Practices in Universities

[2015-02-10] Functional Streams in Lisp Explained

[2015-02-05] Collatz-type Conjectures (Continued)

[2015-02-04] Collatz-type Conjectures (Continued)

[2015-01-31] Collatz-type Conjectures (Continued)

[2015-01-30] Collatz-type Conjectures

[2015-01-28] Experiments with Infinite Recursive Sequences (continued)

[2015-01-17] Experiments with Infinite Recursive Sequences

[2015-01-10] Goldbach Pairs

[2015-01-02] Collatz Lengths (Continued)

[2015-01-01] Functional Streams

[2014-12-27] Polarization in the US Congress

[2014-12-18] Partition a sequence

[2014-11-28] Uniformly Random Permutations

[2014-11-22] An Implementation of Ford-Fulkerson Algorithm in Common Lisp

[2014-11-17] Tropic Calculation of Cheapest Paths

[2014-11-05] Longest common subsequence of two sequences

[2014-10-30] Counting Spanning Trees of a Graph

[2014-10-26] Longest Increasing Subsequence

[2014-10-24] The Number of Inversions in a Sequence

[2014-10-22] Hashes and Entropy

[2014-10-09] Estimating Cardinality with Constant Memory Complexity

[2014-09-30] Landau's Function

[2014-09-29] A Problem on Substitution Ciphers and Group Theory

[2014-09-28] A Morse Code Translator

[2014-09-23] A Memoization Macro for Common Lisp

[2014-09-21] Reducers are Monoid Morphisms

[2014-09-18] Number of isomorphism classes of binary trees

[2014-09-07] CONS is your friend

[2014-08-22] A Zipf's Law Simulation

[2014-08-07] Generating Uniformly Random Trees

[2014-07-09] A Solution for Project Euler #463

[2014-06-12] Entropy of truncated MD5 hashing

[2014-06-08] Hexadecimal digits of π

[2014-02-11] Information content of n-grams

[2014-02-08] Turkish Sentiment Analysis Using Thesaurus Distance

[2014-02-01] Sentiment analysis using word distances

[2014-01-27] Phase transitions in entropy

[2013-12-13] Optimal length of n-grams

[2013-12-10] Counting strings that contain intervals of same letter repetitions

[2013-12-02] Patterns Separating Large Texts

[2013-11-23] Collatz Sequences (Continued)

[2013-11-11] Entropy and approximately one-to-one maps

[2013-10-23] Tree Isomorphism

[2013-10-15] Self Organizing Maps

[2013-09-15] Euler Project #401

[2013-09-15] An additively recursive definition of the Moebius function

[2013-09-11] An Unsuccessful Attempt for Solving Euler Project #401

[2013-09-04] Uniform Sampling from Parametrized Submanifolds in Scala

[2013-09-04] Uniform Sampling from Parametrized Submanifolds

[2013-08-30] Randomly Generated Points Obeying a Distribution

[2013-08-25] Simulated Annealing in Lisp

[2013-08-21] Eigenvalues and Eigenvectors in GSLL

[2013-08-16] Reservoir Sampling

[2013-08-11] Gibbs sampling in lisp compared with C

[2013-08-10] Logistic Regression in lisp

[2013-08-10] Linear Discriminant Analysis in R

[2013-07-17] A Gradient Descent Implementation in Lisp

[2013-07-01] k-Nearest Neighbor Classification Algorithm Implemented in Lisp

[2013-05-19] Newton-Raphson Method

[2013-05-07] Levenshtein Distance

[2013-04-15] Cut points in a graph

[2013-04-01] Experiments on Collatz Lengths (Continued)

[2013-02-18] The sound of the torsion parts of homotopy groups of spheres

[2013-02-12] Monadic Units

[2013-02-07] Distribution of Collatz Lengths (continued)

[2013-02-03] Distribution of Collatz Lengths

[2013-01-31] Quotients of polynomial algebras

[2013-01-12] Path ideals

[2013-01-10] McCarthy91 Terminates

[2013-01-09] Finding all paths in a directed graph

[2013-01-04] A Simple Monte-Carlo Integration Implementation in Lisp

[2012-12-30] A simple problem in Kolmogorov-Chaitin complexity

[2012-12-29] From walks to paths

[2012-12-16] Higher order functions, functors and monads

[2012-12-13] Eccentricity, Radius and Diameter in an Undirected Graph

[2012-11-29] Untitled

[2012-11-25] Strictly Increasing Labels of Directed Graphs

[2012-11-19] Strictly Increasing Labellings of Directed Graphs

[2012-11-17] Nilpotent elements in an artinian algebra

[2012-11-04] Local rings, idempotents and non-invertible elements

[2012-10-18] An implementation of the fixed-radius near neighbor clustering algorithm in lisp

[2012-10-15] Reducing directed graphs

[2012-10-10] An implementation of the k-means clustering algorithm in lisp

[2012-10-08] A comparison of different map functions in lisp

[2012-10-03] Source code entropy

[2012-09-28] Collisions in random walks

[2012-09-26] Transitive closure of a directed graph

[2012-09-26] Solving linear equations in ℕ

[2012-09-26] Listing partitions

[2012-09-26] Inverting formal power series

[2012-09-26] Hasse subgraph of a directed graph