A048670 Jacobsthal function A048669 applied to the product of the first n primes (A002110).

2, 4, 6, 10, 14, 22, 26, 34, 40, 46, 58, 66, 74, 90, 100, 106, 118, 132, 152, 174, 190, 200, 216, 234, 258, 264, 282, 300, 312, 330, 354, 378, 388, 414, 432, 450, 476, 492, 510, 538, 550, 574, 600, 616, 642, 660, 686, 718, 742, 762, 798, 810, 834, 858, 876, 908, 926, 954

Offset: 1

Author

Jan Kristian Haugland

Keywords

• nonn
• nice
• hard

Comments

Pintz shows that j(x#) >= (2*e^gamma + o(1)) x log x log log log x / (log log x)^2 and hence a(n) >= (2*e^gamma + o(1)) n log^2 n log log log n / (log log n)^2 by the Prime Number Theorem. - Charles R Greathouse IV, Sep 08 2012

Jacobsthal conjectures that a(n) >= j(k) := A048669(k) for any k with n prime factors, which would make this the RECORDS transform of A048669. Hajdu & Saradha disprove the conjecture, showing that this fails for n = 24 where j(k) = 236 > 234 = a(24) for any k divisible by 76964283982898776138308824190 and with 24 prime factors in total. - Charles R Greathouse IV, Sep 08 2012 / Edited by Jan Kristian Haugland, Feb 02 2019

Ford, Green, Konyagin, Maynard, & Tao show that j(x#) >> x log x log log log x / log log x and hence a(n) >> n log^2 n log log log n / log log n. - Charles R Greathouse IV, Mar 29 2018

Computation of a(62)-a(64) was supported by Google Cloud. - Andrzej Bozek, Mar 14 2021

References

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

Links

• Andrzej Bozek, Table of n, a(n) for n = 1..64
• Andrzej Bozek, Gerbicz's table with added a(58)-a(64)
• Fintan Costello and Paul Watts, A computational upper bound on Jacobsthal's function, arXiv:1208.5342 [math.NT], 2012.
• Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and Terence Tao, Long gaps between primes, arXiv:1412.5029 [math.NT], 2014-2016; Journal of the American Mathematical Society 31:1 (2018), pp. 65-105.
• Robert Gerbicz, Table of n, a(n), u(n) for n=1..57, where every integer from [u(n),u(n)+a(n)-2] is divisible by at least one of the first n primes. Note that u(n) is not unique, the smallest is given by A049300(n).
• Thomas R. Hagedorn, Computation of Jacobsthal's function h(n) for n < 50, Math. Comp. 78 (2009) 1073-1087.
• L. Hajdu and N. Saradha, Disproof of a conjecture of Jacobsthal, Mathematics of Computation 81 (2012), pp. 2461-2471.
• H. Iwaniec, On the error term in the linear sieve, Acta Arithmetica 19 (1971), pp. 1-30.
• Helmut Maier and Carl Pomerance, Unusually large gaps between consecutive primes, Transactions of the American Mathematical Society 322:1 (1990), pp. 201-237.
• János Pintz, Very large gaps between consecutive primes, Journal of Number Theory 63 (1997), pp. 286-301.
• Stéphane Vinatier and Reg Alcorn, Mathématiques et Art, Gaz. Soc. Math. de France (in French, 2024) No. 181, 44-58. See p. 57.
• Stéphane Vinatier and William Reginald Alcorn, Mathematics as seen by an artist: Inspiring mathematical objects, Eur. Math. Soc. Mag. (2025) Vol. 135, 20-31. See p. 30.
• Mario Ziller, New computational results on a conjecture of Jacobsthal, arXiv:1903.11973 [math.NT], 2019.
• Mario Ziller, On differences between consecutive numbers coprime to primorials, arXiv:2007.01808 [math.NT], 2020.
• Mario Ziller and John F. Morack, Algorithmic concepts for the computation of Jacobsthal's function, arXiv:1611.03310 [math.NT], 2016.

Crossrefs

Cf. A048669, A002110, A005867, A049300, A058989, A072752, A331118.

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[n_] := a[n] = j[primorial[n]]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 10}] (* Jean-François Alcover, Sep 27 2013, after M. F. Hasler *)

Formula

a(n) = A058989(n) + 1.

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

a(n) >= (2*e^gamma + o(1)) n log^2 n log log log n / (log log n)^2, see Pintz.

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

Maier & Pomerance conjecture that Max_{n <= x} A048669(n) = log(x)*(log log x)^(2+o(1)) which suggests a(n) = n*(log n)^(3+o(1)). - Charles R Greathouse IV, Mar 29 2018

a(n) = largest (or last) term in row n of A331118. - Michael De Vlieger, Dec 11 2020

Extensions

a(21)-a(24) from Max Alekseyev, Apr 09 2006

a(25)-a(49) from Thomas Hagedorn, Feb 21 2007

a(46) corrected (published value in Hagedorn's 2009 Mathematics of Computation article was correct) and a(50)-a(54) added by Mario Ziller, Dec 08 2016

a(55)-a(57) from Robert Gerbicz, Apr 10 2017

a(58)-a(64) from Andrzej Bozek, Mar 14 2021