J. Ramanujan Math. Soc. 32, No.4 (2017) 417–430

On the number of factorizations of an integer R. Balasubramanian1,2 and Priyamvad Srivastav1,2 1 Institute

of Mathematical Sciences, Taramani, Chennai, India 600 113 e-mail: [email protected] 2 Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai, India 400 094 e-mail: [email protected]

Communicated by: Prof. Ritabrata Munshi Received: September 25, 2016 Abstract. Let f (n) denote the number of unordered factorizations of a positive integer n into factors larger than 1. We show that the number of f (n), less than or equal to x, is at most  of distinct values  √ log x exp C log log x (1 + o(1)) , where C = 2π 2/3 and x is sufficiently large. This improves upon a previous result of the first author and F. Luca. 2010 Mathematics Subject Classification: Primary: 11A51, 05A99; Secondary: 11B73.

1. Introduction Let f (n) denote the number of unordered factorizations of n into factors larger than 1. More precisely, f (n) is the number of tuples (n 1 , . . . , n r ), such that 1 < n 1 ≤ n 2 ≤ · · · ≤ n r and n = n 1 n 2 . . . n r . For example, f (18) = 4, since 18 has the factorizations 18,

2 · 9,

3 · 6,

2 · 3 · 3.

The function f (n) is a multiplicative analogue of the the partition function. There are various results on the properties of this function. The problem of determining the exact nature of f (n) was considered by Oppenheim [Opp]. He proved that   x exp(2 log x) f (n) ∼ √ . (1.1) 2 π(log x)3/4 n≤x 417