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The article focuses on a property of positive integers depicting the interpedendence between their construction and the form of their prime factors as established by E. A. Karatsuba and M. E. Changa. Karatsuba has discovered that the deviation of the cardinalities of sets of numbers with an odd and an even number of prime factors has the tendency to reach infinity under certain conditions on the factors. It presents several theorems which constitute the distribution of positive integers whose prime factors belong to certain subsets.

Let p denote a prime and P 2 denote an almost prime with at most two prime factors. The author proves that for sufficiently large x, [Figure not available: see fulltext.] where the constant 1.13 constitutes an improvement of the previous result 1.104 due to J. Wu. [ABSTRACT FROM AUTHOR]

The main result of this paper is the fact that the fraction of primes p ≤ x satisfying the condition that p − 1 has a prime divisor q > exp(ln x/ln ln x) and the number of prime divisors of q − 1 essentially differ from ln ln( x/n), where n = ( p − 1)/ q, tends to zero as x increases. [ABSTRACT FROM AUTHOR]

For a finite group G, let τ( G) denote a set of primes such that a prime p belongs to τ( G) if and only if p is a divisor of the index of some maximal subgroup of G. It is proved that if G satisfies any one of the following conditions: (1) G has a p-complement for each p∈τ( G); (2) ∣τ( G)∣=2; (3) the normalizer of a Sylow p-subgroup of G has prime power index for each odd prime p∈τ( G); then G either is solvable or G/Sol( G)≅(2, 7) where Sol( G) is the largest solvable normal subgroup of ( G). [ABSTRACT FROM AUTHOR]

We introduce a new class of lightlike submanifolds, namely, Screen Cauchy Riemann (SCR) lightlike submanifolds of indefinite Kaehler manifolds. Contrary to CR-lightlike submanifolds, we show that SCR-lightlike submanifolds include invariant (complex) and screen real subcases of lightlike submanifolds. We study some properties of proper totally umbilical SCR-lightlike submanifolds, their invariant (complex) and screen real subcases. [ABSTRACT FROM AUTHOR]

Let d[sub k](n) denote the k-fold iterated divisor function (k ≥ 2). It is proved that for sufficiently large x, d[sub k](n) = d[sub k](n + 1) holds for ≫ x(log log x)[sup -3] integers n ≤ x. [ABSTRACT FROM AUTHOR]