stapled intervals (following A090318)

[Max Alekseyev]

SeqFans,

I'm about to submit the following sequences you may find entertaining.

A090318 defines stapled sequence as an interval of positive integers
that does not contain an element coprime to every other element of the
interval. In other word, a sequence 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.
A090318 gives the smallest stapled interval of the given length n>=17.

In particular, 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
that 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.

Sequence of starting points of stapled intervals is:
2184, 27828, 27829, 27830, 32214, 57860, 62244, 87890, 92274, 110990,
117920, 122304, 127374, 147950, 151058, 151059, 151060, 151061,
151062, 152334, 163488, 171054, 177980, 182364, 185924, 185925,
185926, 208010, 212394, 238040, 242424, 249678, 260810, 260811,
260812, 260813, 260814, 264498, 268070, 272454, 298100, 302484,
320870, 320871, 320872, 323510, 324564, 328130, 332514, 339434,
339435, 339436, 339437, 339438, 347004, 358160, 362544, 388190,
392574, 399500, 409188, 409189, 409190, 418220, 422600, 422601,
422602, 422603, 422604, 448250, 452634, 471014, 471015, 471016,
478280, 482664

Call a stapled interval "maximum" if it is not a proper sub-interval
of any other stapled interval. Starting points of maximum stapled
intervals are:
2184, 27828, 32214, 57860, 62244, 87890, 92274, 110990, 117920,
122304, 127374, 147950, 151058, 152334, 163488, 171054, 177980,
182364, 185924, 208010, 212394, 238040, 242424, 249678, 260810,
264498, 268070, 272454, 298100, 302484, 320870,  323510, 324564,
328130, 332514, 339434, 347004, 358160, 362544, 388190, 392574,
399500, 409188, 418220, 422600, 448250, 452634, 471014, 478280, 482664

Similarly, call a stapled interval "minimum" if it does not contain
any stapled proper subinterval. Starting points of minimum stapled
intervals are:
2184, 27830, 32214, 57860, 62244, 87890, 92274, 110990, 117920,
122304, 127374, 147950, 151062, 152334, 163488, 171054, 177980,
182364, 185926, 208010, 212394, 238040, 242424, 249678, 260814,
264498, 268070, 272454, 298100, 302484, 320872, 323510, 324564,
328130, 332514, 339438, 347004, 358160, 362544, 388190, 392574,
399500, 409190, 418220, 422604, 448250, 452634, 471016, 478280, 482664

Max