A048669 The Jacobsthal function g(n): maximal gap in a list of all the integers relatively prime to n.

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 3, 2, 2, 4, 2, 4, 3, 4, 2, 4, 2, 4, 2, 4, 2, 6, 2, 2, 3, 4, 3, 4, 2, 4, 3, 4, 2, 6, 2, 4, 3, 4, 2, 4, 2, 4, 3, 4, 2, 4, 3, 4, 3, 4, 2, 6, 2, 4, 3, 2, 3, 6, 2, 4, 3, 6, 2, 4, 2, 4, 3, 4, 3, 6, 2, 4, 2, 4, 2, 6, 3, 4, 3, 4, 2, 6, 3, 4, 3, 4, 3, 4, 2, 4, 3, 4, 2, 6, 2, 4, 5

Offset: 1

Author

Jan Kristian Haugland

Keywords

• nonn
• easy
• nice

Comments

Equivalently, g(n) is the least integer such that among any g(n) consecutive integers i, i+1, ..., i+g(n)-1 there is at least one which is relatively prime to n.

The definition refers to all integers, not just those in the range 1..n-1.

Differs from A070194 by 1 at the primes. - T. D. Noe, Mar 21 2007

Jacobsthal's function is used in the proofs of Recamán's and Pomerance's conjectures on P-integers--see A192224. - Jonathan Sondow, Jun 14 2014

Examples

			g(6)=4 because the gap between 1 and 5, both being relatively prime to 6, is maximal and 5-1 = 4.
g(7)=2, because the numbers relatively prime to 7 are 1,2,3,4,5,6,8,9,10,..., and the biggest gap is 2. Similarly a(p) = 2 for any prime p. - _N. J. A. Sloane_, Sep 08 2012
		

References

• E. Jacobsthal, Uber Sequenzen ganzer Zahlen, von denen keine zu n teilerfremd ist, I, II, III. Norske Vid. Selsk. Forh., 33, 1960, 117-139.
• D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Pages 33-34.
• E. Westzynthius, Uber die Verteilung der Zahlen, die zu der n ersten Primzahlen teilerfremd sind, Comm. Phys. Math. Helsingfors 25 (1931), 1-37.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Fintan Costello, and Paul Watts, A short note on Jacobsthal's function, arXiv preprint arXiv:1306.1064 [math.NT], 2013.
• P. Erdős, On the integers relatively prime to n and on a number theoretic function considered by Jacobsthal. Math. Scand., 10, 1962, 163-170.
• H. Iwaniec, On the problem of Jacobsthal, Demonstratio Math. 11 (1978), pp. 225-231.
• Hans-Joachim Kanold, Über eine zahlentheoretische Funktion von Jacobsthal, Mathematische Annalen 170.4 (1967): 314-326.
• Gerhard R. Paseman, Updating an upper bound of Erik Westzynthius, arXiv preprint arXiv:1311.5944 [math.NT], 2013-2014.
• Carl Pomerance, A note on the least prime in an arithmetic progression, Journal of Number Theory 12.2 (1980): 218-223.
• Harlan Stevens, On Jacobsthal's g(n)-function, Mathematische Annalen 226.1 (1977): 95-97.
• Mario Ziller, John F. Morack, Algorithmic concepts for the computation of Jacobsthal's function, arXiv:1611.03310 [math.NT], 2016.

Crossrefs

Essentially same as A049298. See A132468 for another version.

Cf. A048670, A070971, A007947, A038566, A192224, A285183.

Programs

Haskell

a048669 n = maximum $ zipWith (-) (tail ts) ts where
   ts = a038566_row n ++ [n + 1]
-- Reinhard Zumkeller, Oct 01 2012

Mathematica

g[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]; Table[g[n], {n, 1, 105}] (* Jean-François Alcover, Sep 03 2013, after M. F. Hasler *)
Table[Max[Differences[Select[Range[110],CoprimeQ[#,n]&]]],{n,110}] (* Harvey P. Dale, Jan 10 2022 *)

PARI

A048669(n)=my(L=1,m=1);for(k=2,n+1,gcd(k,n)>1 && next;L+mM. F. Hasler, Sep 08 2012

Formula

From N. J. A. Sloane, Apr 19 2017 (Start):

g(n) = g(Rad(n)) (cf. A007947). So in studying g(n) we may focus on the case when n is a product of w (say) distinct primes.

g(n) <= 2^w for all w [Kanold].

g(n) <= 2^(1/w) for all w >= e^50 [Kanold].

For some unknown X, g(n) <= X*(w*log(w))^2 for all w [Iwaniec].

(End)

g(n) << (log(n))^2, as proved by Iwaniec. - Charles R Greathouse IV, Sep 08 2012.

Extensions

Edited, changed symbol to g(n), added references pertaining to bounds. - N. J. A. Sloane, Apr 19 2017