2018 
Eggleton RB, Kimberley JS, MacDougall JA, 'Squarefree rank of integers', Journal of Combinatorial Mathematics and Combinatorial Computing, 106 185207 (2018)
© 2018 Charles Babbage Research Centre. All rights reserved. For any positive integer n, there are integers a, b such that a2 is the largest square divisor of n, and n = a2b: the ... [more]
© 2018 Charles Babbage Research Centre. All rights reserved. For any positive integer n, there are integers a, b such that a2 is the largest square divisor of n, and n = a2b: the squarefree rank of n is the number of prime divisors of 6. For instance, 242 = 112.2 and 243 = 92.3 lie in a run of 5 consecutive integers of squarefree rank 1. Below 1010, the longest run of integers with constant squarefree rank has 20 members, all of squarefree rank 3. For each k = 1, we give an upper bound on the size of runs of integers of squarefree rank k, and show (at least for k = 7) there are infinitely many pairs of consecutive integers of squarefree rank k. For distinct primes p, q there is a smallest positive integer d(p, q) such that 0 < a2pb2q = d(p, q) = p holds for infinitely many integers a, b. We study sequences (a1,a2,... ,ak) for rank 1 integers 2a21 > 3a22 > · · · > pka2k with pi(ai + 1)2 > 2a21for 1 < i < k, and the corresponding sequences in which all the inequalities are reversed and {equation presented}. We show that (3n2  n ¿ N) has no member in the interval [2a2,2(a+ 1)2] with a ¿ N if and only if a is in the Beatty sequence ([n (3 + v6)]  n ¿ N); several generalisations are studied. Finally, we study closest linear combinations {equation presented}.



2012 
MacDougall JA, Sweet LG, 'Rational homogeneous algebras', Canadian Mathematical BulletinBulletin Canadien De Mathematiques, 55 351354 (2012) [C1]



2012 
MacDonald GW, MacDougall JA, Sweet LG, 'On the dimension of linear spaces of nilpotent matrices', Linear Algebra and Its Applications, 436 22102230 (2012) [C1]



2012 
Gray ID, MacDougall JA, 'Vertexmagic labeling of regular graphs: Disjoint unions and assemblages', Discrete Applied Mathematics, 160 11141125 (2012) [C1]



2012 
Kimberley JS, MacDougall JA, 'Mutation of vertexmagic regular graphs', Journal of Combinatorial Mathematics and Combinatorial Computing, 82 157177 (2012) [C1]



2012 
Eggleton RB, Kimberley JS, MacDougall JA, 'Runs of integers with equally many distinct prime factors', Bulletin of the Institute of Combinatorics and its Applications, 64 3038 (2012) [C1] 


2011 
Kimberley JS, MacDougall JA, 'All regular graphs of small odd order are vertexmagic', Australasian Journal of Combinatorics, 51 175199 (2011) [C1]



2010 
Mohr SH, MacDougall JA, 'Integral Trees of Diameter 4', AKCE International Journal of Graphs and Combinatorics, 7 171188 (2010) [C1] 


2010 
Gray ID, MacDougall JA, 'Vertexmagic labelling of nonregular graphs', Australasian Journal of Combinatorics, 46 173183 (2010) [C1]



2009 
Gray ID, MacDougall JA, 'Vertexmagic labelings of regular graphs II', Discrete Mathematics, 309 59865999 (2009) [C1]



2009 
Sweet LG, MacDougall JA, 'The maximum dimension of a subspace of nilpotent matrices of index 2', Linear Algebra and Its Applications, 431 11161124 (2009) [C1]



2009 
Adams P, Eggleton RB, MacDougall JA, 'Graphs which are linked cycles', Congressus Numerantium, 195 7583 (2009) [C1] 


2009 
Eggleton RB, Adams P, MacDougall JA, 'Graphs which are linked structures', Congressus Numerantium, 199 7588 (2009) [C1] 


2009 
Gray ID, MacDougall JA, 'Vertexmagic labelings: Mutations', Australasian Journal of Combinatorics, 45 189206 (2009) [C1]



2008 
MacDougall JA, Wallis WD, 'Strong edgemagic graphs of maximum size', Discrete Mathematics, 308 27562763 (2008) [C1]



2008 
Chisholm CR, MacDougall JA, 'Rational tetrahedra with edges in geometric progression', Journal of Number Theory, 128 251262 (2008) [C1]



2008 
Buchholz RH, MacDougall JA, 'Cyclic polygons with rational sides and area', Journal of Number Theory, 128 1748 (2008) [C1]



2008 
Eggleston RB, MacDougall JA, 'Consecutive integers with equally many principal divisors', Mathematics Magazine, 81 235248 (2008) [C1] 


2006 
Chisholm CR, MacDougall JA, 'Rational and Heron tetrahedra', Journal of Number Theory, 121 153185 (2006) [C1]



2006 
Gray ID, MacDougall JA, 'Sparse semimagic squares and vertexmagic labelings', Ars Combinatoria, 80 225242 (2006) [C1]



2006 
Adams PD, Eggleton RG, MacDougall JA, 'Graphs with a given degree sequence', Congressus Numerantium, 179 1531 (2006) [C1] 


2006 
Adams PD, Eggleton RG, MacDougall JA, 'Taxonomy of Graphs of Order 10', Congressus Numerantium, 180 6580 (2006) [C1] 


2006 
Miller M, Baca M, MacDougall JA, 'Vertexmagic Total Labeling of Generalized Petersen Graphs and Convex Polytopes', Journal of Combinatorial Mathematics and Combinatorial Computing, 59 8999 (2006) [C1] 


2006 
Gray ID, MacDougall JA, 'Sparse antimagic squares and vertexmagic labelings of bipartite graphs', Discrete Mathematics, 306 28782892 (2006) [C1]



2005 
Chisholm CR, MacDougall JA, 'Rational tetrahedra with edges in arithmetic progression', Journal of Number Theory, 111 5780 (2005) [C1]



2004 
MacDougall JA, 'A generalisation of the Copair Construction for trianglesaturated graphs', Utilitas Mathematica, 66 239247 (2004) [C1] 


2004 
Adams P, Eggleton RB, MacDougall JA, 'Structure of graph posets for orders 4 to 8', Congressus Numerantium, 166 6381 (2004) [C1] 


2004 
Adams P, Eggleton RB, MacDougall JA, 'Degree sequences and poset structure of order 9 graphs', Congressus Numerantuium, 166 8395 (2004) [C1] 


2003 
MacDougall JA, McSorley JP, Wallis W, Gray ID, 'Vertexmagic total labelings of trees and forests', Discrete Mathematics, 261 285298 (2003) [C1]



2003 
MacDougall JA, 'Heron Triangles with Sides in Arithmetic Progression', Journal of Recreational Mathematics, 31 189196 (2003) [C1] 


2003 
MacDougall JA, 'Mersenne Composites and Cyclotomic Primes', Mathematical Gazette, 87 7175 (2003) [C1] 


2003 
Daven M, MacDougall JA, Rodger C, 'Maximal Sets of Hamilton Cycles in Complete Multipartite Graphs', Journal of Graph Theory, 43 4966 (2003) [C1]



2003 
Gray ID, MacDougall JA, Simpson RJ, Wallis WD, 'VertexMagic Total Labelings of Complete Bipartite Graphs', Ars Combinatoria, 69 117127 (2003) [C1]



2003 
MacDougall JA, Wallis WD, 'Strong edgemagic labelling of a cycle with a chord', Australasian Journal of Combinatorics, 28 245255 (2003) [C1]



2003 
Eggleton RB, MacDougall JA, 'Graph Posets, SpanningUniversal Graphs and Graph Coverings', Bulletin of the ICA, 37 6372 (2003) [C1] 


2003 
Adams P, Eggleton RB, MacDougall JA, Mahmoodian ES, 'Corrections to Steinbach's Posets of Graphs (Orders 5, 6, 7)', Bulletin of the ICA, 37 2934 (2003) [C1] 


2003 
Gray I, MacDougall JA, Wallis WD, 'On VertexMagic Labeling of Complete Graphs', Bulletin of the ICA, 38 4244 (2003) [C1] 


2003 
Baca M, Bertault FA, MacDougall JA, Miller M, Simanjuntak RMG, Slamin, 'VertexAntimagic total labelings of graphs', Discussiones Mathematicae Graph Theory, 23 6783 (2003) [C1] 


2002 
MacDougall JA, Miller M, Wallis W, Slamin, 'VertexMagic Total Labelings of Graphs', Utilitas Mathematica, 61 321 (2002) [C1]



2002 
Eggleton RB, MacDougall JA, 'Minimally trianglesaturated graphs: adjoining a single vertex', Australasian Journal of Combinatorics, 25 263278 (2002) [C1]



2002 
Sweet LG, MacDougall JA, 'A decomposition theorem for homogeneous algebras', Journal of the Australian Mathematical Society, 72 4756 (2002) [C1]



2002 
MacDougall JA, Miller M, Wallis WD, 'Vertexmagic total labelings of wheels and related graphs', Utilitas Mathematica, 62 175183 (2002) [C1]



2001 
MacDougall JA, 'W.D. Wallis, Magic Graphs', Australian Mathematical Society Gazette, 28 275276 (2001) [C3] 


2001 
Eggleton RB, MacDougall JA, 'Minimally StarSaturated Graphs', Congressus Numerantium, 149 161176 (2001) [C1] 


2001 
MacDougall JA, 'Midhat Gazale, Number: From Ahmes to Cantor', The Physicist, 38 1 (2001) [C3] 


2000 
MacDougall JA, Buchholz RH, 'When Newton met Diophantus: A Study of RationalDerived Polynomials and Their Extension to Quadratic Fields', Journal of Number Theory, 81 210233 (2000) [C1]



2000 
Baca M, MacDougall JA, Miller M, Wallis WD, Slamin, 'Survey of certain valuations of graphs', Discussiones Mathematicae: Graph Theory, 20 219229 (2000) [C1] 


1999 
Buchholz RH, MacDougall JA, 'Heron Quadrilaterals with sides in arithmetic or geometric progression', Bull. Austral. Math. Soc., 59 263269 (1999) [C1]



1999 
Boswell S, Eggleton RB, MacDougall JA, 'Minimally pathsaturated graphs', Congressus Numerantium, 138 (1999) 97117 (1999) [C1] 


1999 
MacDougall JA, Mullen GL, 'A Note on BIB Designs with Nested Rows and Columns', Australasian Journal of Combinatorics, 20 257259 (1999) [C1] 


1998 
Eggleton RB, MacDougall JA, 'Almost $K_m$saturated graphs', Congressus Numerantium, 131 187203 (1998) [C1] 


1988 
SWEET LG, MACDOUGALL JA, 'ON AUTOMORPHISMS OF ORDER3 OF DIVISIONALGEBRAS', LINEAR ALGEBRA AND ITS APPLICATIONS, 108 231237 (1988)


