

A194585


Starting points of stapled intervals of length 17.


3



2184, 27830, 32214, 57860, 62244, 87890, 92274, 117920, 122304, 147950, 152334, 177980, 182364, 208010, 212394, 238040, 242424, 268070, 272454, 298100, 302484, 328130, 332514, 358160, 362544, 388190, 392574, 418220, 422604, 448250
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OFFSET

1,1


COMMENTS

"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 [am,bm] 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 or equal to ba+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)


LINKS

Fidel I. Schaposnik, Table of n, a(n) for n = 1..666


FORMULA

From Colin Barker, Aug 16 2014: (Start)
a(n) = (15031+10631*(1)^n+30030*n)/2.
a(n) = a(n1)+a(n2)a(n3).
G.f.: 2*x*(1100*x^2+12823*x+1092) / ((x1)^2*(x+1)). (End)


PROG

(PARI) {u=vector(17, j, 1); v=vector(17, j, j); for(k=2, 1e9, nextprime(k)<k+17 & (v+=u*(v[1]+k=precprime(k+17))) & next; v+=u; for(j=k, k+16, vecsort(gcd(j, v), , 8)[2]<j  next(2)); print1(k", "))}


CROSSREFS

Cf. A090318, A130170, A130171, A130173.
Sequence in context: A130170 A090318 A130171 * A107657 A002521 A059757
Adjacent sequences: A194582 A194583 A194584 * A194586 A194587 A194588


KEYWORD

nonn


AUTHOR

M. F. Hasler, Oct 14 2011


STATUS

approved



