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A244620 Initial terms of Erdős-Wood intervals of length 22.

%I A244620 #25 Jul 04 2023 11:43:19
%S A244620 3521210,6178458,13220900,15878148,22920590,25577838,32620280,
%T A244620 35277528,42319970,44977218,52019660,54676908,61719350,64376598,
%U A244620 71419040,74076288,81118730,83775978,90818420,93475668,100518110,103175358,110217800,112875048,119917490
%N A244620 Initial terms of Erdős-Wood intervals of length 22.
%C A244620 By definition of the intervals in A059756, these are numbers that start a sequence of 23 consecutive integers such that none of the 23 integers is coprime to the first and also coprime to the last integer of the interval.
%C A244620 Hence each initial term of an Erdős-Wood interval is the initial term of a stapled interval of length A059756(n) + 1 (see definition in A090318). - _Christopher Hunt Gribble_, Dec 02 2014
%H A244620 Christopher Hunt Gribble, <a href="/A244620/b244620.txt">Table of n, a(n) for n = 1..1000</a>
%H A244620 Wikipedia, <a href="http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Woods_number">Erdős-Woods number</a>
%H A244620 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, -1).
%F A244620 a(1) = A059757(2).
%F A244620 From _Christopher Hunt Gribble_, Dec 02 2014: (Start)
%F A244620 a(1) = A130173(524).
%F A244620 a(2*n+1) = 3521210 + 9699690*n.
%F A244620 a(2*n+2) = 6178458 + 9699690*n.
%F A244620 a(n) = (-4849867 - 2192597*(-1)^n + 9699690*n)/2.
%F A244620 a(n) = a(n-1) + a(n-2) - a(n-3).
%F A244620 G.f.: (3521232*x^2+2657248*x+3521210) / ((x-1)^2*(x+1)). (End)
%e A244620 3521210 = 2*5*7*11*17*269 and 3521210+22 = 3521232 = 2^4 * 3^4 * 11 * 13 * 19, and all numbers in [3521210,3521232] have at least one prime factor in {2, 3, 5, 7, 11, 13, 17, 19, 269}. Therefore 3521210 is in the list.
%p A244620 isEWood := proc(n,ewlength)
%p A244620     local nend,fsn,fsne,fsall,fsk ;
%p A244620     nend := n+ewlength ;
%p A244620     fsn := numtheory[factorset](n) ;
%p A244620     fsne := numtheory[factorset](nend) ;
%p A244620     fsall := fsn union fsne ;
%p A244620     for k from n to nend do
%p A244620         fsk := numtheory[factorset](k) ;
%p A244620         if fsk intersect fsall = {} then
%p A244620             return false;
%p A244620         end if;
%p A244620     end do:
%p A244620     return true;
%p A244620 end proc:
%p A244620 for n from 2 do
%p A244620     if isEWood(n,22) then
%p A244620         print(n) ;
%p A244620     end if;
%p A244620 end do:
%Y A244620 Cf. A059757, A194585, A090318, A130173.
%K A244620 nonn
%O A244620 1,1
%A A244620 _R. J. Mathar_, Jul 02 2014
%E A244620 More terms from _Christopher Hunt Gribble_, Dec 03 2014