A194585 -id:A194585 - cp's OEIS frontend

A059757 Initial terms of smallest Erdős-Woods intervals corresponding to the terms of A059756.

2184, 3521210, 47563752566, 12913165320, 21653939146794, 172481165966593120, 808852298577787631376, 91307018384081053554, 1172783000213391981960, 26214699169906862478864, 27070317575988954996883440, 92274830076590427944007586984, 3061406404565905778785058155412

Offset: 1

Nik Lygeros (webmaster(AT)lygeros.org), Feb 12 2001

nonn

The term a(1)=2184 is the start of the lowest interval with an Erdős-Woods interval length of 16 = A059756(1), and the list of others of that length 16 appear to be the same as A194585. - R. J. Mathar, Jul 02 2014

			For the Erdős-Woods number 16, no number between 2184 and 2184+16 is coprime to both 2184 and 2184+16. 2184 is the smallest such nonnegative integer.

Bertram Felgenhauer, Table of n, a(n) for n = 1..106
Code Golf Stack Exchange user xash, Find the Erdős-Woods origin.
Bertram Felgenhauer, Some OEIS computations (first 106 terms and supporting code)

Cf. A059756, A194585, A244620.

a(5)-a(13) corrected by Peter Kagey, Jul 01 2021, based on the Code Golf Stack Exchange link.

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

Original entry on oeis.org

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

R. J. Mathar, Jul 02 2014

nonn

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

			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.

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).

Cf. A059757, A194585, A090318, A130173.

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:

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)

More terms from Christopher Hunt Gribble, Dec 03 2014

cp's OEIS Frontend

A059757 Initial terms of smallest Erdős-Woods intervals corresponding to the terms of A059756.

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