

A059756


ErdősWoods numbers: the length of an interval of consecutive integers with property that every element has a factor in common with one of the endpoints.


9



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
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OFFSET

1,1


COMMENTS

"Length" means total number of terms including endpoints, 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 p1n1 or p2n2.  M. F. Hasler, Jun 29 2014; there is now a proof for this (see Gribble link), Dec 17 2014
In popular culture: this sequence was involved in the encryption of a message in Episode "eps2.9_pyth0npt1.p7z" of the "Mr. Robot" TV series (first aired Sep 14 2016).  Jessica K. Sklar, Jan 30 2019


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) 5362.
D. Dowe, On the existence of sequences of coprime pairs of integers, J. Austral. Math. Soc. Ser. A 47, no. 1, 8489. 1989
P. Erdős and J. L. Selfridge, Complete prime subsets of consecutive integers, Proceedings of the Manitoba Conference on Numerical Mathematics, 1971, pp. 114.
M. F. Hasler, R. J. Mathar, Corrigendum to "Paths and circutis in finite groups", Discr. Math. 22 (1978) 263, arXiv:1510.07997 (2015).
Christopher Hunt Gribble, Seqfan thread, Dec 05 2014
W. Holsztynski, R. F. E. Strube, Paths and circuits in finite groups, Discr. Math. 22 (1978) 263272.
Internet Movie Database, Mr. Robot: Season 2, Episode 11: eps2.9_pyth0npt1.p7z
Nik Lygeros, ErdosWoods Numbers
W. T. Trotter and Paul Erdős, When the Cartesian product of directed cycles is Hamiltonian, J. Graph Theory 2 (1978) 137142
Wikipedia, ErdosWoods number
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(n1))); forstep(x1=2, 2^#P1, 2, P1=vecextract(P, x1); P2=setminus(P, P1); for(n1=1, n1, bittest(nn1, 0)  next; setintersect(P1, factor(n1)[, 1]~)  setintersect(P2, factor(nn1)[, 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



