A060680 Smallest difference between consecutive divisors of n.

1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 2, 1, 22, 1, 4, 1, 2, 1, 28, 1, 30, 1, 2, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 2, 1, 46, 1, 6, 1, 2, 1, 52, 1, 4, 1, 2, 1, 58, 1, 60, 1, 2, 1, 4, 1, 66, 1, 2, 1, 70, 1, 72, 1, 2, 1, 4, 1, 78, 1, 2, 1, 82, 1, 4, 1, 2, 1, 88, 1, 6, 1, 2, 1, 4

Offset: 2

Author

Labos Elemer, Apr 19 2001

Keywords

• nonn

Comments

a(n) = 1 if n is even and a(n) is even if n is odd.

a(n) = least m>0 such that n!+1+m and n-m are not relatively prime. - Clark Kimberling, Jul 21 2012

Examples

			For n = 35, divisors = {1,5,7,35}; differences = {4,2,28}; a(35) = smallest difference = 2.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 2..10000
• Antal Balog, Paul Erdős and Gérald Tenenbaum, On Arithmetic Functions Involving Consecutive Divisors, In: Analytical Number Theory, pp. 77-90, Birkhäuser, Basel, 1990.

Crossrefs

Cf. A060681 (largest difference), A060682, A060683, A060684.

Cf. A193829, A027750.

Programs

Haskell

a060680 = minimum . a193829_row  -- Reinhard Zumkeller, Jun 25 2015

Maple

read("transforms") :
A060680 := proc(n)
    sort(convert(numtheory[divisors](n),list)) ;
    DIFF(%) ;
    min(op(%)) ;
end proc:
seq(A060680(n),n=2..60) ; # R. J. Mathar, May 23 2018

Mathematica

a[n_] := Min@@(Drop[d=Divisors[n], 1]-Drop[d, -1]);
(* Second program: *)
a[n_] := Min[Differences[Divisors[n]]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Oct 16 2024 *)

PARI

a(n) = {my(m = n, d1); fordiv(n, d, if(d > 1 && d - d1 < m, m = d - d1); d1 = d); m;} \\ Amiram Eldar, Mar 17 2025

Formula

a(2n+1) = A060684(n).

Extensions

Corrected by David W. Wilson, May 04 2001

Edited by Dean Hickerson, Jan 22 2002