Bi-unitary multiperfect numbers, IV(c)

Abstract

A divisor d of a positive integer n is called a unitary divisor if gcd(d, n/d) = 1; and d is called a bi-unitary divisor of n if the greatest common unitary divisor of d and nld is unity. The concept of a bi-unitary divisor is due to D. Surynarayana (1972). Let sigma** (n) denote the sum of the bi-unitary divisors of n. A positive integer n is called a bi-unitary multiperfect number if sigma** (n) = kn for some k >= 3. For k = 3 we obtain the bi-unitary triperfect numbers. Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers. The present paper is part IV(c) in a series of papers on even bi-unitary multiperfect numbers. In parts I, II and III we determined all bi-unitary triperfect numbers of the form n = 2(a)u, where 1 <= a <= 6 and u is odd. In part V we fixed the case a = 8. The case a = 7 is more difficult. In Parts IV(a-b) we solved partly this case, and in the present paper (Part IV(c)) we continue the study of the same case (a = 7).