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%I #31 Jul 03 2023 14:25:13
%S 2184,27830,32214,57860,62244,87890,92274,117920,122304,147950,152334,
%T 177980,182364,208010,212394,238040,242424,268070,272454,298100,
%U 302484,328130,332514,358160,362544,388190,392574,418220,422604,448250
%N Starting points of stapled intervals of length 17.
%C "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.
%C From _Fidel I. Schaposnik_, Aug 16 2014: (Start)
%C 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.
%C Then it is clear that if [a,b] is a stapled interval, so is [m+a,m+b].
%C 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".
%C 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.
%C 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].
%C The g.f. encompasses both these chains, namely a(2*n+1) = 2184 + 30030*n and a(2*n+2) = 27830 + 30030*n.
%C (End)
%H Fidel I. Schaposnik, <a href="/A194585/b194585.txt">Table of n, a(n) for n = 1..666</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, -1).
%F From _Colin Barker_, Aug 16 2014: (Start)
%F a(n) = (-15031 + 10631*(-1)^n + 30030*n)/2.
%F a(n) = a(n-1) + a(n-2) - a(n-3).
%F G.f.: 2*x*(1100*x^2 + 12823*x + 1092) / ((x-1)^2*(x+1)). (End)
%o (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","))}
%Y Cf. A090318, A130170, A130171, A130173.
%K nonn
%O 1,1
%A _M. F. Hasler_, Oct 14 2011
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