Note on a theorem of Zehnxiag Zhang

A sequence of strictly positive integers is said to be primitive if none of its terms divides the others. In this paper, we give a new proof of a result, conjectured by P. Erdős and Z. Zhang in 1993, on a primitive sequence whose the number of the prime factors of the termes counted with multiplicity is at most 4. The objective of this proof is to improve the complexity, which helps to prove this conjecture.

On a translated sum over primitive sequences related to a conjecture of Erdős

For x large enough, there exists a primitive sequence \mathcal{A}, such that \begin{equation*} \sum\limits_{a\in \mathcal{A}}\frac{1}{a(\log a+x)}\gg \sum\limits_{p\in \mathcal{P}}\frac{1}{p(\log p+x)}\text{,} \end{equation*} where \mathcal{P} denotes the set of prime numbers.

Complex hybridity in Isotoma petraea. III. Lethal system in Θ12 Bencubbin

Analysis of selfed capsules of a ring of 12 (Θ12) complex heterozygote confirms the presence of a balanced lethal system. One of the complexes, S' is two interchanges removed from the primitive chromosome end sequence and when homozygous, causes the abortion of seeds after differentiation of the testa has begun. The other complex, Ne, is three interchanges removed from the primitive sequence and the lethals in it operate, when homozygous, before the testa differentiates. Twin hybrids from crosses between the Θ12 and an alethal structural homozygote exhibit epistatic segregation ratios in their selfed capsules, demonstrating that the lethals in the two complexes are not controlled by single loci. An hypothesis based on relatively translocated internal chromosome segments is developed as the basis for the evolutionary elaboration of the balanced lethal systems in I. Petraea complex hybrids.

Primitive sequences whose elements have no large prime factors

A sequence a1 < a2 < … of positive integers is said to be primitive if no element of the sequence divides any other. The study of primitive sequences arose naturally out of investigations into the subject of abundant numbers, where sequences each of whose elements is of the form , the pi being fixed primes, are of particular importance. Such a sequence is said to be built up from the primes p1…pr. Thus Dickson [1], in an early paper on abundant numbers, proved that a primitive sequence built up from a fixed set of primes is necessarily finite.

On a theorem of Behrend

A sequence of integers 0 <a1 < a2 <… no term of which divides any other will be called a primitive sequence. Throughout this paper c1, c2,… will denote suitable positive absolute constants. Behrend [1] proved that for every primitive sequence