A conjecture of De Koninck regarding particular square values of the sum of divisors function
Broughan, K. A., Delbourgo, D., & Zhou, Q. (2014). A conjecture of De Koninck regarding particular square values of the sum of divisors function. Journal of Number Theory, 137, 50–66. https://doi.org/10.1016/j.jnt.2013.10.011
Permanent Research Commons link: https://hdl.handle.net/10289/11326
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. If the prime dividing a solution n is congruent to 3 modulo 8 then it must be greater than 41, and every solution is divisible by at least the fourth power of an odd prime. Moreover at least 2/5 of the exponents a of the primes dividing any solution have the property that a + 1 is a prime power. Lastly we prove that the number of solutions up to x > 1 is at most x¹/⁶⁺є, for any є > 0 and all x > xє.
This is an author’s accepted version of an article published in the journal: Journal of Number Theory. © 2013 Elsevier Inc.