Journal of Integer Sequences, Vol. 15 (2012), Article 12.7.5

On Integers for Which the Sum of Divisors is the Square of the Squarefree Core

Kevin A. Broughan
Department of Mathematics
University of Waikato
Private Bag 3105
Hamilton, New Zealand

Jean-Marie De Koninck
Départment de mathématiques et de statistique
Université Laval
Québec G1V 0A6

Imre Kátai
Department of Computer Algebra
Pázmány Péter sétány I/C
H-1117 Budapest

Florian Luca
Centro de Ciencias Matemáticas
Universidad Nacional Autonoma de México
C. P. 58089
Morelia, Michoacán


We study integers n > 1 satisfying the relation σ(n) = γ(n)2, where σ(n) and γ(n) are the sum of divisors and the product of distinct primes dividing n, respectively. We show that the only solution n with at most four distinct prime factors is n = 1782. We show that there is no solution which is fourth power free. We also show that the number of solutions up to x > 1 is at most x1/4+ε for any ε > 0 and all x > xε. Further, call n primitive if no proper unitary divisor d of n satisfies σ(d) | γ(n)2. We show that the number of primitive solutions to the equation up to x is less than xε for x > xε.

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Received February 23 2012; revised versions received July 26 2012; August 14 2012; September 3 2012. Published in Journal of Integer Sequences, September 8 2012.

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