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%I #24 Feb 26 2023 18:53:53
%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.
%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