On the ratio of the sum of divisors and Euler’s totient function II

We find the form of all solutions to ø(n) | σ(n) with three or fewer prime factors, except when the quotient is 4 and n is even

Divisibility by 3 of even multiperfect numbers of abundancy 3 and 4

We say a number is flat if it can be written as a non-trivial power of 2 times an odd squarefree number. The power is the “exponent” and the number of odd primes the “length”. Let N be flat and 4-perfect with exponent a and length m. If a ≢ 1 mod 12, then a is even. If a is even and 3 ∤ N then m is also even. If a ≡ 1 mod 12 then 3 | N and m is even. If N is flat and 3-perfect and 3 ∤ N, then if a a ≡ 1 mod 12, a is even. If a ≡ 1 mod 12 then m is odd. If N is flat and 3 or 4-perfect then it is divisible by at least one Mersenne prime, but not all odd prime divisors are Mersenne. We also give some conditions for the divisibility by 3 of an arbitrary even 4-perfect number

Odd multiperfect numbers of abundancy 4

Euler's structure theorem for any odd perfect number is extended to odd multiperfect numbers of abundancy power of 2. In addition, conditions are found for classes of odd numbers not to be 4-perfect: some types of cube, some numbers divisible by 9 as the maximum power of 3, and numbers where 2 is the maximum even prime power

Flat primes and thin primes

A number is called upper (lower) flat if its shift by +1 ( −1) is a power of 2 times a squarefree number. If the squarefree number is 1 or a single odd prime then the original number is called upper (lower) thin. Upper flat numbers which are primes arise in the study of multi-perfect numbers. Here we show that the lower or upper flat primes have asymptotic density relative to that of the full set of primes given by twice Artin’s constant, that more than 53% of the primes are both lower and upper flat, and that the series of reciprocals of the lower or the upper thin primes converges

Multiply perfect numbers of low abundancy

The purpose of this thesis is to investigate the properties of multiperfect numbers with low abundancy, and to include the structure, bounds, and density of certain multiperfect numbers. As a significant result of this thesis, an exploration of the structure of an odd 4-perfect number has been made. An extension of Euler’s theorem on the structure of any odd perfect number to odd 2k-perfect numbers has also been obtained. In order to study multiperfect numbers, it is necessary to discuss the factorization of the sum of divisors, in particular for (qe), for prime q. This concept is applied to investigate multiperfect numbers with a so-called flat shape N = 2ap1 · · ·pm. If some prime divisors of N are fixed then there are finitely many flat even 3-perfect numbers. If N is a flat 4-perfect number and the exponent of 2 is not congruent to 1 (mod 12), then the exponent is even. If all odd prime divisors of N are Mersenne primes, where N is even, flat and multiperfect, then N is a perfect number. In more general cases, some necessary conditions for the divisibility by 3 of an even 4-perfect number N = 2ab are obtained, where b is an odd positive integer. Two new ideas, namely flat primes and thin primes, are introduced since these appear often in multiperfect numbers. The relative density of flat primes to all primes is given by 2 times Artin’s constant. An upper bound of the number of thin primes is T(x) less less x log2 x . The sum of the reciprocals of the thin primes is finite

The Stability Analysis of Foundation Pit Under Seepage State Based on Plaxis Software

The stability of excavation engineering is closely related to groundwater, so it is important to study the impact of seepage flow on the stability of foundation pit. The work is based on the percolation theory and principles of strength reduction. The computation were done with use of the Plaxis software. There were studied simulations which included the seepage state and simulations which didn\u27t include this effect. In order to studythe influence of seepage on the stability of foundation pit, there was computed the stability coefficient by using the strength. The results show that, when the seepage stability was not considered, the coefficient is 30% larger than when considering the seepage. Therefore, when designing and calculating the excavation, the seepage should be considered when checking stability if there is groundwater