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We establish upper bounds for the six smallest prime factors of odd perfect, quasiperfect, and odd almost perfect numbers.

Abstract. We prove that an odd triperfect number has at least ten distinct prime factors. The following lemmas are easy to prove:Lemma l.IfNisOT,(1) a/s are even for 1 < I < r.Lemma 2. // TV is OT and q is a prime factor of a(pf-) for some i, then q = 3 or q = pj for some j, 1 ^ y < r.Lemma 3. If N is OT and r = 9, ps < 80.

Hagis and McDaniel have shown that the largest prime factor of an odd perfect number TV is at least 100111, and Pomerance has shown that the second largest prime factor is at least 139. Using these facts together with the method we develop, we show that if 3 | N, N is divisible by at least ten distinct primes.

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A simple, highly extensible computational strategy to assess compound toxicity has been developed with the premise that a compound's toxicity can be gauged from the toxicities of structurally similar compounds. Using a reference set of 13645 compounds with reported acute toxicity endpoint dose data (oral, rat-LD(50) data normalized in mg/kg), a generic utility which assigns a compound the average toxicity of structurally similar compounds is shown to correlate well with reported values. In a leave-one-out simulation using the requirement that at least one structurally similar member in a "voting consortium" is present within a reference set, the strategy demonstrates a predictive correlation (q wedge 2) of 0.82 with 57.3% of the compounds being predicted. Similar leave-one-out simulations on a set of 1781 drugs demonstrate a q wedge 2 of 0.74 with 51.8% of the compounds being predicted. Simulations to optimize similarity cutoff definitions, consortium member size, and reference set size illustrate that a significant improvement in the quality and quantity of predictions can be obtained by increasing the reference set size. Similar application of the strategy to subchronic and chronic toxicity data should be possible given reasonably sized reference sets.

Abstract.We make a table of odd integers N with five distinct prime factors for which 2 -10-12 < a(N)/N < 2 + 10-12, and show that for such N \a(N)/N -2 I > 10-14.Using this inequality, we prove that there are no odd perfect numbers, no quasiperfect numbers and no odd almost perfect numbers with five distinct prime factors. We also make a table of odd primitive abundant numbers N with five distinct prime factors for which 2 < a(N)/N < 2 + 2/1010.

Abstract. We prove that odd perfect numbers not divisible by 3 have at least eleven distinct prime factors.1. N is called a perfect number if a(7V) = 27V, where a(TV) is the sum of positive divisors of TV. Twenty-seven even perfect numbers are known; however, no odd perfect (OP) numbers have been found.Suppose TV is OP and 6. Pomerance (1972, [7]) and Robbins (1972) proved that «(TV) > 1. Hagis (1975, [2]) and Chein (1978, [1]) proved that 100129, and Pomerance [8] proved that the second largest prime factor of TV > 139.If 31 TV, then Kanold (1949) proved that co(TV) s= 9, and the author (1977, [4]) proved that to(TV) > 10.In this paper we prove Theorem. If N is OP and3\N, then 11.2. In the remainder of this paper we assume that TV is OP and