Krystal Hirsch

Extremal and Average Behavior of a Shifted GCD Sum over Divisors

For every integer n >= 2, define A(n) = sum_{d|n} gcd(d-1, n/d-1), and write B(n) = A(n) - 2(n-1). We prove the sharp inequality A(n) <= 9n/4, with equality only at n = 36, and determine the next two extremal ratios: after excluding 36, equality in A(n) <= 20n/9 occurs only at n...

Extremal and Average Behavior of a Shifted GCD Sum over Divisors

For every integer n >= 2, define A(n) = sum_{d|n} gcd(d-1, n/d-1), and write B(n) = A(n) - 2(n-1). We prove the sharp inequality A(n) <= 9n/4, with equality only at n = 36, and determine the next two extremal ratios: after excluding 36, equality in A(n) <= 20n/9 occurs only at n...