A090318 -id:A090318 - cp's OEIS frontend

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.

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.

A130173 Starting points of stapled intervals.

Original entry on oeis.org

2184, 27828, 27829, 27830, 32214, 57860, 62244, 87890, 92274, 110990, 117920, 122304, 127374, 147950, 151058, 151059, 151060, 151061, 151062, 152334, 163488, 171054, 177980, 182364, 185924, 185925, 185926, 208010, 212394

Offset: 1

Max Alekseyev, Jul 24 2007

A finite sequence of n consecutive positive integers is called "stapled" if each element in the sequence is not relatively prime to at least one other element in the sequence.
In other words, an interval is stapled if for every element x there is another element 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;
* for any particular positive integer a, we can determine if it is a starting point of some stapled interval.

H. L. Nelson, There is a better sequence, Journal of Recreational Mathematics, Vol. 8(1), 1975, pp. 39-43.

Fidel I. Schaposnik, Table of n, a(n) for n = 1..1492 (first 76 terms from Max Alekseyev)
A. Brauer, On a Property of k Consecutive Integers, Bull. Amer. Math. Society, vol. 47, 1941, pp. 328-331.
R. J. Evans, On Blocks of N Consecutive Integers, Amer. Math. Monthly, vol. 76, 1969, pp. 48-49.
Irene Gassko, Stapled Sequences and Stapling Coverings of Natural Numbers, Electronic Journal of Combinatorics, Vol. 3, 1996, Paper R33.

Cf. A090318, A130170, A130171.

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

A152816 Minimum positive initial term of a minimum length n-fold stapled sequence.

Original entry on oeis.org

1, 2184, 2496060719312

Offset: 0

William Rex Marshall, Dec 13 2008

In an n-fold stapled sequence, each term in the sequence is not coprime to at least n other terms. The minimum length of an n-fold stapled sequence is A152817(n). The known terms are also the minimum possible for an n-fold stapled sequence of any length.

			For n = 2, the 41 consecutive integers of the first minimum length 2-fold stapled sequence is given in the following list along with the prime factors each term shares with at least two other terms:
2496060719312: 2, 17
2496060719313: 3, 7
2496060719314: 2, 19, 37
2496060719315: 5
2496060719316: 2, 3
2496060719317: 11
2496060719318: 2
2496060719319: 3
2496060719320: 2, 5, 7, 31
2496060719321: 13
2496060719322: 2, 3
2496060719323: 23, 29
2496060719324: 2
2496060719325: 3, 5
2496060719326: 2
2496060719327: 7
2496060719328: 2, 3, 11
2496060719329: 17
2496060719330: 2, 5
2496060719331: 3
2496060719332: 2
2496060719333: 19
2496060719334: 2, 3, 7, 13
2496060719335: 5
2496060719336: 2
2496060719337: 3
2496060719338: 2
2496060719339: 11
2496060719340: 2, 3, 5
2496060719341: 7
2496060719342: 2
2496060719343: 3
2496060719344: 2
2496060719345: 5
2496060719346: 2, 3, 17, 23
2496060719347: 13
2496060719348: 2, 7
2496060719349: 3
2496060719350: 2, 5, 11
2496060719351: 31, 37
2496060719352: 2, 3, 19, 29

Cf. A090318, A152817 (minimum length).

cp's OEIS Frontend

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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A130173 Starting points of 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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A152816 Minimum positive initial term of a minimum length n-fold stapled sequence.

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