GENERALISED QUASIPERFECT NUMBERS. (WITH AN APPENDIX ON ODD PERFECT NUMBERS)

The main concern of this thesis is an investigation of the equation (sigma)(n) = 2n + k('2), where n and k are relatively prime positive integers, k is odd and (sigma)(n) is the sum of the positive divisors of n. We have called any n satisfying this equation for some k a generalised quasiperfect number, or, specifically, a k-quasiperfect number, since such n, for k = 1, are quasiperfect numbers. We know of no example, for any k, nor can we prove for any k that none exist. Nonetheless, we have established a great number of necessary conditions for the existence of a generalised quasiperfect number.

Some of the main results are as follows. Suppose there is a k-quasiperfect number n. Then n is an odd perfect square, has at lest four prime factors, and exceeds 10('20). If n has only four or five prime factors, then k exceeds 10('10). If k (LESSTHEQ) 44366047, then n is primitive abundant. If n is primitive abundant, then n has at least six prime factors.

The thesis includes also a more general investigation of primitive (alpha)-abundant numbers. We show that, for positive integers x, y, z, if x(sigma)(n) = yn + z, n is not primitive (y/x)-abundant and n/s is not (y/x)-perfect (where s('d) is the largest prime power in n), then

n < max{2(z - 1)('5/2)/y, 4(z + 1/2)('3)/27y}.

We table all solutions of the equation (sigma)(n) = yn + z(y (GREATERTHEQ) 2, z (LESSTHEQ) 210) for which n is not primitive y-abundant. Some new properties of a primitive (alpha)-abundant number n are obtained: (sigma)(n)/n < (alpha) + min{ 1/2, 3(alpha)e('-5(alpha)/9) /2} and (sigma)(n)/n < (alpha) + 1.6(alpha)/log n. We list all primitive abundant numbers n with (sigma)(n)/n (GREATERTHEQ) 2.05 and show that (sigma)(n)/n < 2.16104.

The first result of the preceding paragraph has a useful application to generalised quasiperfect numbers. We use it in showing that, if n is k-quasiperfect and not primitive abundant, then n < 2(1 + 10('-15))k('6)/27. This makes use of a result developed in connection with the existence of odd perfect numbers.

It was conjectured in 1975 that the equation (sigma)(n) = 2n + k('2), with k odd, has no solutions at all. We show this is not true, and exhibit seven solutions. The final chapter of the thesis considers this and other aspects of the equation (sigma)(n) = 2n + k('2) when some of our original restrictions on n and k are relaxed.

In the Appendix, we produce two recently submitted papers which give new results on the possible exponents that the prime factors of an odd perfect number might have. If

(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)

is the prime factor decomposition of an odd perfect number, then we show that

(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)