Bi-unitary multiperfect numbers, IV(b)

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 n/d is unity. The concept of a bi-unitary divisor is due to D. Surynarayana (1972). Let \sig^{**}(n) denote the sum of the bi-unitary divisors of n. A positive integer n is called a bi-unitary multiperfect number if \sig^{**}(n)=kn for some k\geq 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(b) in a series of papers on even bi-unitary multiperfect numbers. In parts I, II and III we considered bi-unitary triperfect numbers of the form n=2^{a}u, where 1\leq a \leq 6 and u is odd. In part IV(a) we solved partly the case a=7. We proved that if n is a bi-unitary triperfect number of the form n=2^{7}.5^{b}.17^{c}.v, where (v, 2.5.17)=1, then b\geq 2. We then solved completely the case b=2. In the present paper we give some partial results concerning the case b\ge 3 under the assumption 3\nmid n.

Bi-unitary multiperfect numbers, IV(c)

“…We assume that the reader has Parts I, II, III, IV(a-b), V (see [2][3][4][5][6][7]) available. We, however, recall Lemmas 2.1-2.3 from these parts because they are so important also here.…”

“…We have σ * * (19) = 20 = 2 2 .5. Taking f = 1 in (3.7b), it follows that its right hand side is divisible by 2 5 and its left hand side is unitarily divisible by 2 6 . Hence w ′ is 1 or an odd prime power.…”

Bi-unitary multiperfect numbers, V

“…We assume that the reader has parts I, II, III, IV(a-b) (see [2][3][4][5][6]) available. We, however, recall Lemma 2.1 from these parts because it is so important also here.…”

“…In parts I, II and III (see [2][3][4]) we considered bi-unitary triperfect numbers of the form n = 2 a u, where 1 ≤ a ≤ 6 and u is odd. In parts IV(a-b) (see [5,6]) we solved partly the case a = 7. In this paper we fix the case a = 8.…”

“…11.13 e−1 .76 .61 g−1 .w = 19.1201.σ * * (13 e ).σ * * (61 g ).σ * * (w ). (3.14d) From (3.14b), w is divisible by 19 and 1201.…”