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

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

Canada

Imre Kátai

Department of Computer Algebra

Pázmány Péter sétány I/C

H-1117 Budapest

Hungary

Florian Luca

Centro de Ciencias Matemáticas

Universidad Nacional Autonoma de México

C. P. 58089

Morelia, Michoacán

México

**Abstract:**

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 *x*^{1/4+ε}
for any ε > 0 and all
*x* > *x*_{ε}. 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
*x*^{ε}
for *x* > *x*_{ε}.

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.

Return to