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

Author

William Rex Marshall, Jan 25 2004

Keywords

• nonn
• nice

Comments

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

Examples

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

References

• 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.

Links

• 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.

Crossrefs

Cf. A130170, A130171, A130173.

Programs

Mathematica

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 *)

Extensions

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