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

3521210, 6178458, 13220900, 15878148, 22920590, 25577838, 32620280, 35277528, 42319970, 44977218, 52019660, 54676908, 61719350, 64376598, 71419040, 74076288, 81118730, 83775978, 90818420, 93475668, 100518110, 103175358, 110217800, 112875048, 119917490

Offset: 1

Author

R. J. Mathar, Jul 02 2014

Keywords

• nonn

Comments

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.

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

Examples

			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.
		

Links

• Christopher Hunt Gribble, Table of n, a(n) for n = 1..1000
• Wikipedia, Erdős-Woods number
• Index entries for linear recurrences with constant coefficients, signature (1, 1, -1).

Crossrefs

Cf. A059757, A194585, A090318, A130173.

Programs

Maple

isEWood := proc(n,ewlength)
    local nend,fsn,fsne,fsall,fsk ;
    nend := n+ewlength ;
    fsn := numtheory[factorset](n) ;
    fsne := numtheory[factorset](nend) ;
    fsall := fsn union fsne ;
    for k from n to nend do
        fsk := numtheory[factorset](k) ;
        if fsk intersect fsall = {} then
            return false;
        end if;
    end do:
    return true;
end proc:
for n from 2 do
    if isEWood(n,22) then
        print(n) ;
    end if;
end do:

Formula

a(1) = A059757(2).

From Christopher Hunt Gribble, Dec 02 2014: (Start)

a(1) = A130173(524).

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

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

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

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

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

Extensions

More terms from Christopher Hunt Gribble, Dec 03 2014