On Highly Factorable Numbers Original Research Article

For a positive integern, letf(n) be the number of multiplicative partions ofn. We say that a nutural numbernis highly factorable iff(m)ajthenf(npj/pi)greater-or-equal, slantedf(n). Using this fact, we prove the conjecture of Canfield, Erdimages, and Pormerance: for each fixedk, ifnis a large highly factorable number then there are asymptotically exactly 1/k(k+1) of the exponents ofnwhich are equal tok. We also answer the questions posed by Canfieldet al.: ifn,n′ are consecutive highly factorable numbers, then does it follown′/n→1 andf(n′)/f(n)→1