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A finite sequence of n consecutive positive integers is called "stapled" if each term in the sequence is not relatively prime to at least one other term in the sequence. Thus a staple joins two terms of the sequence whose gcd is > 1.

It has been proved that stapled intervals of length n >= 17 exist for all n.

From Max Alekseyev, Jul 24 2007: (Start)

An interval is stapled if for every term x there is another term y (different from x) such that gcd(x,y) > 1.

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

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

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).

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

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

* we can determine whether any particular positive integer a is a starting point of some stapled interval. (End)

For n >= 17, a(n) < A034386(n-1) = (n-1)#. - Max Alekseyev, Oct 08 2007

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.

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.

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

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

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).

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

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

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

"Stapled" intervals are defined in A090318. They are at least of length 17, and those of this minimal length are listed here. Therefore, this is not only a subsequence of A130173, but also of A130171.

From Fidel I. Schaposnik, Aug 16 2014: (Start)

Let S be the set of distinct prime factors appearing in the factorization of at least two different numbers in the range [a,b], and m the product of all the elements in S.

Then it is clear that if [a,b] is a stapled interval, so is [m+a,m+b].

Moreover, if a > m then the range [a-m,b-m] is also a stapled interval of the same length, so we can group the stapled intervals of a given length in "chains".

To prove the g.f., note that S cannot contain any prime number greater than or equal to b-a+1, so for stapled intervals of length 17 the maximum value of m is m = 2*3*5*7*11*13 = 30030.

Then any stapled interval of length 17 must belong to a chain whose first element is at most 30030, and the only stapled intervals in this range are [2184,2200] and [27830,27846].

The g.f. encompasses both these chains, namely a(2*n+1) = 2184 + 30030*n and a(2*n+2) = 27830 + 30030*n.

(End)