A244620 - OEIS

%I #25 Jul 04 2023 11:43:19

%S 3521210,6178458,13220900,15878148,22920590,25577838,32620280,

%T 35277528,42319970,44977218,52019660,54676908,61719350,64376598,

%U 71419040,74076288,81118730,83775978,90818420,93475668,100518110,103175358,110217800,112875048,119917490

%N Initial terms of Erdős-Wood intervals of length 22.

%C 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 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 Christopher Hunt Gribble, <a href="/A244620/b244620.txt">Table of n, a(n) for n = 1..1000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Woods_number">Erdős-Woods number</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, -1).

%F a(1) = A059757(2).

%F From _Christopher Hunt Gribble_, Dec 02 2014: (Start)

%F a(1) = A130173(524).

%F a(2*n+1) = 3521210 + 9699690*n.

%F a(2*n+2) = 6178458 + 9699690*n.

%F a(n) = (-4849867 - 2192597*(-1)^n + 9699690*n)/2.

%F a(n) = a(n-1) + a(n-2) - a(n-3).

%F G.f.: (3521232*x^2+2657248*x+3521210) / ((x-1)^2*(x+1)). (End)

%e 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 isEWood := proc(n,ewlength)

%p     local nend,fsn,fsne,fsall,fsk ;

%p     nend := n+ewlength ;

%p     fsn := numtheory[factorset](n) ;

%p     fsne := numtheory[factorset](nend) ;

%p     fsall := fsn union fsne ;

%p     for k from n to nend do

%p         fsk := numtheory[factorset](k) ;

%p         if fsk intersect fsall = {} then

%p             return false;

%p         end if;

%p     end do:

%p     return true;

%p end proc:

%p for n from 2 do

%p     if isEWood(n,22) then

%p         print(n) ;

%p     end if;

%p end do:

%Y Cf. A059757, A194585, A090318, A130173.

%K nonn

%O 1,1

%A _R. J. Mathar_, Jul 02 2014

%E More terms from _Christopher Hunt Gribble_, Dec 03 2014