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A058989 Largest number of consecutive integers such that each is divisible by a prime <= the n-th prime. 4
1, 3, 5, 9, 13, 21, 25, 33, 39, 45, 57, 65, 73, 89, 99, 105, 117, 131, 151, 173, 189, 199, 215, 233, 257, 263, 281, 299, 311, 329, 353, 377, 387, 413, 431, 449, 475, 491, 509, 537, 549, 573, 599, 615, 641, 659, 685, 717, 741, 761, 797, 809, 833, 857 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Marty Weissman conjectured that a(n)=2q-1, where q is the largest prime smaller than the n-th prime. The conjecture holds for the first few terms, but then a(n) is larger than 2q-1. Phil Carmody proved a(n)>=2q-1. Terms were calculated by Weissman, Carmody and McCranie.

A049300(n) is the smallest value of the mentioned consecutive integers. - Reinhard Zumkeller, Jun 14 2003

REFERENCES

Dickson, L. E., History of the Theory of Numbers, Vol. 1, p. 439, Chelsea, 1952.

H. Iwaniec, On the error term in the linear sieve, Acta. Arith. 19 (1971), pp. 1-30.

J. D. Laison and M. Schick, "Seeing dots: visibility of lattice points", Mathematics Magazine, Vol. 80 (2007), pp. 274-282. See page 281 reference 13.

János Pintz, Very large gaps between consecutive primes, Journal of Number Theory 63 (1997), pp. 286-301.

LINKS

Table of n, a(n) for n=1..54.

Thomas R. Hagedorn, Computation of Jacobsthal's function h(n) for n < 50, Math. Comp. 78 (2009) 1073-1087.

Mario Ziller, John F. Morack, Algorithmic concepts for the computation of Jacobsthal's function, arXiv:1611.03310 [math.NT], 2016.

FORMULA

a(n) = A048670(n) - 1. See that entry for additional information.

Iwaniec proved that a(n) << n^2*(log n)^2. - Charles R Greathouse IV, Sep 08 2012

a(n) >= (2e^gamma + o(1)) n log^2 n log log log n / (log log n)^2, see A048670. - Charles R Greathouse IV, Sep 08 2012

a(n) = 2 * A072752(n) + 1. - Mario Ziller, Dec 08 2016

See A048669 for many other bounds and references - N. J. A. Sloane, Apr 19 2017

EXAMPLE

The 4th prime is 7. Nine is the maximum number of consecutive integers such that each is divisible by 2, 3, 5 or 7. (Example: 2 through 10) So a(4)=9.

CROSSREFS

Cf. A048669, A048670, A072752.

Sequence in context: A178415 A249424 A076274 * A049691 A206297 A227565

Adjacent sequences:  A058986 A058987 A058988 * A058990 A058991 A058992

KEYWORD

nice,nonn,changed

AUTHOR

Jud McCranie, Jan 16 2001

EXTENSIONS

Laison and Schick reference from Parthasarathy Nambi, Oct 19 2007

More terms from A048670 added by Max Alekseyev, Feb 07 2008

a(46) corrected and a(50)-a(54) added by Mario Ziller, Dec 08 2016

STATUS

approved

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Last modified April 27 23:18 EDT 2017. Contains 285555 sequences.