

A048670


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


14



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


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


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..61
Andrzej Bozek, Gerbicz's table with added a(58)a(61)
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], 20142016; Journal of the American Mathematical Society 31:1 (2018), pp. 65105.
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.
Thomas R. Hagedorn, Computation of Jacobsthal's function h(n) for n < 50, Math. Comp. 78 (2009) 10731087.
L. Hajdu and N. Saradha, Disproof of a conjecture of Jacobsthal, Mathematics of Computation 81 (2012), pp. 24612471.
H. Iwaniec, On the error term in the linear sieve, Acta Arithmetica 19 (1971), pp. 130.
Helmut Maier and Carl Pomerance, Unusually large gaps between consecutive primes, Transactions of the American Mathematical Society 322:1 (1990), pp. 201237.
János Pintz, Very large gaps between consecutive primes, Journal of Number Theory 63 (1997), pp. 286301.
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, John F. Morack, Algorithmic concepts for the computation of Jacobsthal's function, arXiv:1611.03310 [math.NT], 2016.


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


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}] (* JeanFrançois Alcover, Sep 27 2013, after M. F. Hasler *)


CROSSREFS

Cf. A048669, A002110, A005867, A058989, A072752.
Sequence in context: A023499 A103445 A001747 * A333315 A307889 A239951
Adjacent sequences: A048667 A048668 A048669 * A048671 A048672 A048673


KEYWORD

nonn,nice,hard


AUTHOR

Jan Kristian Haugland


EXTENSIONS

a(21)a(24) from Max Alekseyev, Apr 09 2006
a(25)a(49) from Tom Hagedorn (hagedorn(AT)tcnj.edu), Feb 21 2007
a(46) corrected 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(61) from Andrzej Bozek, Aug 25 2020


STATUS

approved



