A058989 Largest number of consecutive integers such that each is divisible by a prime <= the n-th prime.

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, 875, 907, 925, 953, 977, 1001, 1029, 1057, 1097, 1109

Offset: 1

Author

Jud McCranie, Jan 16 2001

Keywords

• nice
• nonn

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

By Lemma 5 of Maynard, there is a constant C > 0 such that, for any n, there is a prime p <= C*exp(prime(n)) such that q - p >= a(n) where q is the smallest prime larger than p. (Probably C = 7/e^3 = 0.35... is admissible.) - Charles R Greathouse IV, Aug 01 2024

Examples

			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.
		

References

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

Links

• Thomas R. Hagedorn, Computation of Jacobsthal's function h(n) for n < 50, Math. Comp. 78 (2009) 1073-1087.
• 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.
• James Maynard, Gaps between primes, arXiv preprint arXiv:1910.13450 [math.NT], 2019.
• Michael Penn, a nice divisibility problem, YouTube video, 2024.
• János Pintz, Very large gaps between consecutive primes, Journal of Number Theory 63 (1997), pp. 286-301.
• Mario Ziller and John F. Morack, Algorithmic concepts for the computation of Jacobsthal's function, arXiv:1611.03310 [math.NT], 2016.

Crossrefs

Cf. A048669, A048670, A072752.

Programs

Mathematica

(* This program is not suitable to compute more than a few terms *)
primorial[n_] := Product[Prime[k], {k, 1, n}];
j[n_] := Module[{L = 1, m = 1}, For[k = 2, k <= n+1, k++, If[GCD[k, n] == 1, If[L+m < k, m = k-L]; L = k]]; m];
a[1] = 1;
a[n_] := a[n] = j[primorial[n]] - 1;
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 10}] (* Jean-François Alcover, Sep 05 2017 *)

PARI

do(n,P,R)=for(i=1,#R, if(n==R[i], return(do(n+1,P,R)))); if(#P==0, return(n-1)); my(b=0); for(i=1,#P,my(t=do(n+1,setminus(P,[P[i]]), concat(R,Mod(n,P[i])))); if(t>b,b=t)); b
a(n)=do(1,primes(n),[]) \\ Charles R Greathouse IV, Aug 08 2024

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

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

a(55)-a(64) from A048670 added by Constantino Calancha, Aug 05 2023