A130173 -id:A130173 - cp's OEIS frontend

A090318 a(n) = least positive k such that k, k+1, k+2, ..., k+n-1 is a stapled interval of length n, or 0 if no such sequence exists.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2184, 27829, 27828, 87890, 87890, 171054, 171054, 323510, 127374, 323510, 151062, 151062, 151062, 151061, 151060, 151059, 151058, 7106718, 7106718, 7567747, 7567746, 7567745, 7567744, 7567743, 7567742, 48595315, 48595314, 48595313

Offset: 1

William Rex Marshall, Jan 25 2004

A finite sequence of n consecutive positive integers is called "stapled" if each term in the sequence is not relatively prime to at least one other term in the sequence. Thus a staple joins two terms of the sequence whose gcd is > 1.
It has been proved that stapled intervals of length n >= 17 exist for all n.
From Max Alekseyev, Jul 24 2007: (Start)
An interval is stapled if for every term x there is another term y (different from x) such that gcd(x,y) > 1.
The shortest stapled interval has length 17 and starts with the number 2184.
It is interesting to notice that the intervals [27829,27846] and [27828,27846] are stapled while the interval [27828,27845] is not.
It is clear that a stapled interval [a,b] may not contain a prime number greater than b/2 (as such a prime would be coprime to every other element of the interval).
Together with Bertrand's Postulate this implies a > b/2 or b < 2a. And it follows that
* a stapled interval may not contain prime numbers at all;
* we can determine whether any particular positive integer a is a starting point of some stapled interval. (End)
For n >= 17, a(n) < A034386(n-1) = (n-1)#. - Max Alekseyev, Oct 08 2007

			The shortest possible stapled sequence is [2184, 2185, 2186, 2187, 2188, 2189, 2190, 2191, 2192, 2193, 2194, 2195, 2196, 2197, 2198, 2199, 2200].

I. Gassko, Common factor graphs of stapled sequences, Proceedings of the Twenty-eighth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1997). Congr. Numer. 126 (1997), 163-173.
I. Gassko, Stapling and composite coverings of natural numbers, Proceedings of the Twenty-seventh Southeastern International Conference on Combinatorics, Graph Theory and Computing (Baton Rouge, LA, 1996). Congr. Numer. 118 (1996), 109-116.
H. L. Nelson, There is a better sequence, Journal of Recreational Mathematics, Vol. 8(1), 1975, pp. 39-43.

Max Alekseyev and William Rex Marshall, Table of n, a(n) for n = 1..103
Ethan Berkove and Michael Brilleslyper, Subgraphs of Coprime Graphs on Sets of Consecutive Integers, Integers (2022) Vol. 22, #A47, see p. 8.
A. Brauer, On a Property of k Consecutive Integers, Bull. Amer. Math. Society, vol. 47, 1941, pp. 328-331.
R. B. Eggleton, Common factors of integers: A graphic view, Discrete Math. 65 (1987), 141-147.
R. J. Evans, On Blocks of N Consecutive Integers, Amer. Math. Monthly, vol. 76, 1969, pp. 48-49.
R. J. Evans, On N Consecutive Integers in an Arithmetic Progression, Acta Sci. Math. Univ. Szeged, vol. 33, 1972, pp. 295-296.
Irene Gassko, Stapled Sequences and Stapling Coverings of Natural Numbers, Electronic Journal of Combinatorics, Vol. 3, Paper R33.
L. Hajdu and N. Saradha, On a problem of Pillai and its generalizations, Acta Arithmetica 144:4 (2010), pp. 323-347.
Heiko Harborth, Eine Eigenschaft Aufeinanderfolgender Zahlen, Arch. Math. (Basel), vol. 21, 1970, pp. 50-51.
Jyhmin Kuo and Hung-Lin Fu, On Near Relative Prime Number in a Sequence of Positive Integers, Taiwanese J. Math. 14 (1) 123-129, 2010.
S. S. Pillai, On m Consecutive Integers I, Proc. Indian Acad. Sci., Sect. A, vol. 11, 1940, pp. 6-12; II ibid., vol. 11, 1940, pp. 73-80, III ibid, vol. 13, 1941, pp. 530-533; IV Bull. Calcutta Math. Soc. 36, 1944, pp. 99-101.

Cf. A130170, A130171, A130173.

dd = 41; nn = 10^7; Clear[sp, L]; sp[] = 0; L[] = 0; For[ i = 0, i < PrimePi[dd], ++i, p = Prime[i + 1]; For[ n = 0, n < nn + dd, n += p, If[sp[n] == 0, sp[n] = p]]]; Print["init done"]; For[ n = 1, n <= nn, ++n, m = 1; For[ d = 0, d < dd, ++d, s = sp[n + d]; If[s == 0, Break[]]; If[s > d, m = Max[m, d + s]]; If[d >= m && L[d] == 0, L[d] = n]] ]; Reap[For[ i = 1, i <= dd, ++i, Print["a[", i, "] = ", L[i - 1]]; Sow[L[i - 1]]]][[2, 1]] (* Jean-François Alcover, Mar 26 2013, translated and adapted from Max Alekseyev's program *)

Edited by N. J. A. Sloane, Aug 04 2007, Oct 08 2007

A130170 "Stapled" intervals are defined in A090318. Call a stapled interval "maximal" if it is not a proper subinterval of any other stapled interval. Sequence gives starting points of maximal stapled intervals.

Original entry on oeis.org

2184, 27828, 32214, 57860, 62244, 87890, 92274, 110990, 117920, 122304, 127374, 147950, 151058, 152334, 163488, 171054, 177980, 182364, 185924, 208010, 212394, 238040, 242424, 249678, 260810, 264498, 268070, 272454

Offset: 1

Max Alekseyev, Jul 24 2007

nonn

Max Alekseyev and Fidel I. Schaposnik, Table of n, a(n) for n = 1..972 (first 50 terms from Max Alekseyev)

Cf. A090318, A130171, A130173.

A130171 "Stapled" intervals are defined in A090318. Call a stapled interval "minimal" if it does not contain any proper stapled subinterval. Sequence gives starting points of minimal stapled intervals.

Original entry on oeis.org

2184, 27830, 32214, 57860, 62244, 87890, 92274, 110990, 117920, 122304, 127374, 147950, 151062, 152334, 163488, 171054, 177980, 182364, 185926, 208010, 212394, 238040, 242424, 249678, 260814, 264498, 268070, 272454

Offset: 1

Max Alekseyev, Jul 24 2007

nonn

Max Alekseyev and Fidel I. Schaposnik, Table of n, a(n) for n = 1..978 (first 50 terms from Max Alekseyev)

Cf. A090318, A130170, A130173.

A194585 Starting points of stapled intervals of length 17.

Original entry on oeis.org

2184, 27830, 32214, 57860, 62244, 87890, 92274, 117920, 122304, 147950, 152334, 177980, 182364, 208010, 212394, 238040, 242424, 268070, 272454, 298100, 302484, 328130, 332514, 358160, 362544, 388190, 392574, 418220, 422604, 448250

Offset: 1

M. F. Hasler, Oct 14 2011

nonn

"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.
From Fidel I. Schaposnik, Aug 16 2014: (Start)
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.
Then it is clear that if [a,b] is a stapled interval, so is [m+a,m+b].
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".
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.
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].
The g.f. encompasses both these chains, namely a(2*n+1) = 2184 + 30030*n and a(2*n+2) = 27830 + 30030*n.
(End)

Fidel I. Schaposnik, Table of n, a(n) for n = 1..666
Index entries for linear recurrences with constant coefficients, signature (1, 1, -1).

Cf. A090318, A130170, A130171, A130173.

{u=vector(17,j,1);v=vector(17,j,j);for(k=2,1e9, nextprime(k)

From Colin Barker, Aug 16 2014: (Start)
a(n) = (-15031 + 10631*(-1)^n + 30030*n)/2.
a(n) = a(n-1) + a(n-2) - a(n-3).
G.f.: 2*x*(1100*x^2 + 12823*x + 1092) / ((x-1)^2*(x+1)). (End)

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

A090318 a(n) = least positive k such that k, k+1, k+2, ..., k+n-1 is a stapled interval of length n, or 0 if no such sequence exists.

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A130170 "Stapled" intervals are defined in A090318. Call a stapled interval "maximal" if it is not a proper subinterval of any other stapled interval. Sequence gives starting points of maximal stapled intervals.

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A130171 "Stapled" intervals are defined in A090318. Call a stapled interval "minimal" if it does not contain any proper stapled subinterval. Sequence gives starting points of minimal stapled intervals.

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A194585 Starting points of stapled intervals of length 17.

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

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