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A130173 Starting points of stapled intervals. 6


%S 2184,27828,27829,27830,32214,57860,62244,87890,92274,110990,117920,

%T 122304,127374,147950,151058,151059,151060,151061,151062,152334,

%U 163488,171054,177980,182364,185924,185925,185926,208010,212394

%N Starting points of stapled intervals.

%C A finite sequence of n consecutive positive integers is called "stapled" if each element in the sequence is not relatively prime to at least one other element in the sequence.

%C In other words, an interval is stapled if for every element x there is another element y (different from x) such that gcd(x,y)>1.

%C The shortest stapled interval has length 17 and starts with the number 2184.

%C It is interesting to notice that the intervals [27829,27846] and [27828,27846] are stapled while the interval [27828,27845] is not.

%C It is clear that a stapled interval [a,b] may not contain a prime number greater than b/2 (as such a prime would be coprime to every other element of the interval).

%C Together with Bertrand's Postulate this implies a>b/2 or b<2a. And it follows that

%C * a stapled interval may not contain prime numbers at all;

%C * for any particular positive integer a, we can determine if it is a starting point of some stapled interval. (End)

%D H. L. Nelson, There is a better sequence, Journal of Recreational Mathematics, Vol. 8(1), 1975, pp. 39-43.

%H Fidel I. Schaposnik, <a href="/A130173/b130173.txt">Table of n, a(n) for n = 1..1492</a> (first 76 terms from Max Alekseyev)

%H A. Brauer, <a href="http://dx.doi.org/10.1090/S0002-9904-1941-07455-0">On a Property of k Consecutive Integers</a>, Bull. Amer. Math. Society, vol. 47, 1941, pp. 328-331.

%H R. J. Evans, <a href="http://www.jstor.org/stable/2316790">On Blocks of N Consecutive Integers</a>, Amer. Math. Monthly, vol. 76, 1969, pp. 48-49.

%H Irene Gassko, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v3i1r33">Stapled Sequences and Stapling Coverings of Natural Numbers</a>, Electronic Journal of Combinatorics, Vol. 3, 1996, Paper R33.

%Y Cf. A090318, A130170, A130171.

%K nonn,nice

%O 1,1

%A _Max Alekseyev_, Jul 24 2007

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