Results 1  10
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8,418
On certain positive integer sequences
, 2004
"... A survey of recent results in elementary number theory is presented in this paper. Special attention is given to structure and asymptotic properties of certain families of positive integers. In particular, a conjecture on complete sequences of Burr, Erdős, Graham and WenChing Li is amended. ..."
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A survey of recent results in elementary number theory is presented in this paper. Special attention is given to structure and asymptotic properties of certain families of positive integers. In particular, a conjecture on complete sequences of Burr, Erdős, Graham and WenChing Li is amended.
REDUCED ^PARTITIONS OF POSITIVE INTEGERS*
, 1992
"... As a generalization of the equation 0(x) + 0(&) = 0(x + /t), 0partitions and reduced 0partitions and reduced 0partitions of positive integers were considered by Patricia Jones [1]. That is., n = ax + —\at is a 0partition if i> 1 and <j)(n) = §{ax) + '' ' + 0 ( a /)> ..."
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As a generalization of the equation 0(x) + 0(&) = 0(x + /t), 0partitions and reduced 0partitions and reduced 0partitions of positive integers were considered by Patricia Jones [1]. That is., n = ax + —\at is a 0partition if i> 1 and <j)(n) = §{ax) + '' ' + 0 ( a
TwoColorings of Positive Integers
, 2008
"... Let f: {1, 2, 3,...} → {−1, 1} be an arbitrary function. Given a threshold M> 0, we ask two questions: • Do there exist integers a> 0, b ≥ 0, ℓ> 0 such that f(a + b) + f(2a + b) + f(3a + b) + · · · + f(ℓ a + b) > M? • Do there exist integers a> 0, ℓ> 0 such that f(a) + f(2a) ..."
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Cited by 1 (0 self)
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) + f(3a) + · · · + f(ℓ a) > M? The answer to the first question is yes. In words, every twocoloring of the positive integers has unbounded discrepancy, taken over the family of arithmetic progressions. Restricting attention to the subset {1, 2, 3,..., n}, we have [1, 2, 3, 4] c n 1/4 ≤ P (n
SUBSEMiGROUPS OF THE ADDITIVE POSITIVE INTEGERS
"... Many of the attempts to obtain representations for commutative and/or Archimedean semigroups involve using the additive positive integers or subsemigroups of the additive positive integers. In this regard note references [1] , [3] , and [4]. The purpose of this paper is to catalogue the results that ..."
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Many of the attempts to obtain representations for commutative and/or Archimedean semigroups involve using the additive positive integers or subsemigroups of the additive positive integers. In this regard note references [1] , [3] , and [4]. The purpose of this paper is to catalogue the results that
A new family of positive integers
 Ann. Comb
"... Let n,p,k be three positive integers. We prove that the numbers () n k 3F2(1 − k, −p,p−n; 1,1−n; 1) are positive integers which generalize the classical binomial coefficients. We give two generating functions for these integers, and a straightforward application. ..."
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Cited by 5 (4 self)
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Let n,p,k be three positive integers. We prove that the numbers () n k 3F2(1 − k, −p,p−n; 1,1−n; 1) are positive integers which generalize the classical binomial coefficients. We give two generating functions for these integers, and a straightforward application.
On sequences of positive integers
 Acta Arithm
, 1937
"... Let a I, a 2i..., am be any finite set of distinct natural numbers, and let b 1, b 2,... be the sequence formed by all those numbers which are divisible by any of a 1, a 2,..., am. This sequence has a density in the obvious sense and we denote this density by A(a I, a 2i..., am). In fact, I I A(a I, ..."
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Cited by 13 (1 self)
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Let a I, a 2i..., am be any finite set of distinct natural numbers, and let b 1, b 2,... be the sequence formed by all those numbers which are divisible by any of a 1, a 2,..., am. This sequence has a density in the obvious sense and we denote this density by A(a I, a 2i..., am). In fact, I I A(a I, a2,...,a.)=! + a
THE STUDY OF POSITIVE INTEGERS (a,6)
"... A Pset will be defined as a set of positive integers such that if a and b are two distinct elements of this set, ab + 1 is a square. There are many examples of Psets such as [2, 12] or [1, 3, 8, 120] and even formulas such as ..."
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A Pset will be defined as a set of positive integers such that if a and b are two distinct elements of this set, ab + 1 is a square. There are many examples of Psets such as [2, 12] or [1, 3, 8, 120] and even formulas such as
Sieving the positive integers by large primes
, 1988
"... Let Q be a set of primes having relative density 6 among the primes, with 0~6 < 1, and let $(x. y. Q) be the number of positive integers <x that have no prime factors from Q exceeding y. We prove that if yt cc, then r&x, y, Q) w xp6(u), where u = (log x)/(log y), and ps is the continuous ..."
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Cited by 4 (0 self)
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Let Q be a set of primes having relative density 6 among the primes, with 0~6 < 1, and let $(x. y. Q) be the number of positive integers <x that have no prime factors from Q exceeding y. We prove that if yt cc, then r&x, y, Q) w xp6(u), where u = (log x)/(log y), and ps is the continuous
ON THE INTEGER PART OF A POSITIVE INTEGER’S
"... Abstract The main purpose of this paper is using the elementary method and analytic method to study the asymptotic properties of the integer part of the kth root positive integer, and give two interesting asymptotic formulae. ..."
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Abstract The main purpose of this paper is using the elementary method and analytic method to study the asymptotic properties of the integer part of the kth root positive integer, and give two interesting asymptotic formulae.
Results 1  10
of
8,418