A note on Deaconescu’s conjecture

Abstract Hasanalizade [5] studied Deaconescu’s conjecture for positive composite integer n . A positive composite integer n ≥ 4 is said to be a Deaconescu number if S 2 ( n ) | ϕ ( n ) − 1. In this paper, we improve Hasanalizade’s result by proving that a Deaconescu number n must have at least seventeen distinct prime divisors, i.e., ω ( n ) ≥ 17 and must be strictly larger than 5.86 · 10 22 . Further, we prove that if any Deaconescu number n has all prime divisors greater than or equal to 11, then ω ( n ) ≥ p *, where p * is the smallest prime divisor of n and if n ∈ D 3 then all the prime divisors of n must be congruent to 2 modulo 3 and ω ( n ) ≥ 48.