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A059756 Erdos-Woods numbers: the length of an interval of consecutive integers with property that every element has a factor in common with one of the end-points. 7
16, 22, 34, 36, 46, 56, 64, 66, 70, 76, 78, 86, 88, 92, 94, 96, 100, 106, 112, 116, 118, 120, 124, 130, 134, 142, 144, 146, 154, 160, 162, 186, 190, 196, 204, 210, 216, 218, 220, 222, 232, 238, 246, 248, 250, 256, 260, 262, 268, 276, 280, 286, 288, 292, 296, 298, 300, 302, 306, 310, 316, 320, 324, 326, 328, 330, 336, 340, 342, 346, 356, 366, 372, 378, 382, 394, 396, 400, 404, 406, 408, 414, 416, 424, 426, 428, 430 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

"Length" means total number of terms including end-points, minus 1.

Woods was the first to find such numbers, Dowe proved there are infinitely many and Cegielski, Heroult and Richard showed that the set is recursive.

This seems to coincide with prime partitionable numbers in sense of Holsztynski & Strube: n such that there is a partition {P1,P2} of the primes less than n such that for any composition n1+n2=n, there is (p1,p2) in P1 x P2 such that p1|n1 or p2|n2. - M. F. Hasler, Jun 29 2014; there is now a proof for this (see Gribble link), Dec 17 2014

REFERENCES

R. K. Guy: Unsolved Problems in Number Theory, 1981, related to Sections B27, B28, B29.

Konstantin Lakkis, Number Theory [in Greek], Revised edition, 1984.

LINKS

Table of n, a(n) for n=1..87.

P. Cegielski, F. Heroult, D. Richard, On the amplitude of intervals of natural numbers whose every element has a common prime divisor with at least an extremity, Theor. Comp. Sci. 303 (1) (2003) 53-62.

D. Dowe, On the existence of sequences of coprime pairs of integers, J. Austral. Math. Soc. Ser. A 47, no. 1, 84-89. 1989

P. Erdos and J. L. Selfridge, Complete prime subsets of consecutive integers, Proceedings of the Manitobe Conference on Numerical Mathematics. pp. 1-14, 1971

Christopher Hunt Gribble, Seqfan thread, Dec 05 2014

W. Holsztynski, R. F. E. Strube, Paths and circuits in finite groups, Discr. Math. 22 (1978) 263-272.

Nik Lygeros, Erdos-Woods Numbers

W. T. Trotter and Paul Erdos, When the Cartesian product of directed cycles is Hamiltonian, J. Graph Theory 2 (1978) 137-142

Alan Robert Woods, Some Problems in Logic and Number Theory, and their Connections, Thesis, University of Manchester, 1981 (reprinted in New Studies in Weak Arithmetics, Sept 2013, CSLI).

EXAMPLE

a(1) = 16 refers to the interval 2184, 2185, ..., 2200. The end points are 2184 = 2^3 *3 *7 *13 and 2200 = 2^3 *5^2 *11, and each number 2184<=k<=2200 has at least one prime factor in the set {2,3,5,7,11,13}.

PROG

(PARI) prime_part(n)={my(P=primes(primepi(n-1))); forstep(x1=2, 2^#P-1, 2, P1=vecextract(P, x1); P2=setminus(P, P1); for(n1=1, n-1, bittest(n-n1, 0) || next; setintersect(P1, factor(n1)[, 1]~) || setintersect(P2, factor(n-n1)[, 1]~) || next(2)); return([P1, P2]))} \\ M. F. Hasler, Jun 29 2014

CROSSREFS

See A059757 for first terms of corresponding intervals. Cf. A111042.

Sequence in context: A064804 A102944 A058901 * A242214 A242559 A242066

Adjacent sequences:  A059753 A059754 A059755 * A059757 A059758 A059759

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

Further terms from Victor S. Miller, Sep 29 2005

STATUS

approved

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Last modified March 29 09:44 EDT 2015. Contains 255996 sequences.