A060682 Number of distinct differences between consecutive divisors of n (ordered by size).

1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 3, 1, 3, 2, 4, 1, 3, 1, 4, 3, 3, 1, 4, 2, 3, 3, 5, 1, 5, 1, 5, 3, 3, 3, 5, 1, 3, 3, 5, 1, 4, 1, 5, 4, 3, 1, 5, 2, 5, 3, 5, 1, 4, 3, 6, 3, 3, 1, 7, 1, 3, 4, 6, 3, 5, 1, 5, 3, 6, 1, 6, 1, 3, 3, 5, 3, 5, 1, 7, 4, 3, 1, 6, 3, 3, 3, 7, 1, 7, 2, 5, 3, 3, 3, 6, 1, 5, 4, 6, 1, 5, 1, 7, 5, 3

Offset: 2

Author

Labos Elemer, Apr 19 2001

Keywords

• nonn

Comments

Number of all differences for n is d(n)-1 = A000005(n)-1. Increments are not necessarily different, so a(n)<=d(n)-1.

Examples

			For n=70, divisors={1,2,5,7,10,14,35,70}; differences={1,3,2,3,4,21,35}; a(70) = number of distinct differences = 6.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 2..10000
• A. Balog, P. Erdős and G. Tenenbaum, On Arithmetic Functions Involving Consecutive Divisors, In: Analytical Number Theory, pp. 77-90, Birkhäuser, Basel, 1990.
• Jason Earls, Smarandache iterations of the first kind on functions involving divisors and prime factors, in Smarandache Notions Journal (2004), Vol. 14.1, page 264.

Crossrefs

Cf. A000005, A060680, A060681, A060683.

Cf. A193829.

Programs

Haskell

import Data.List (nub, genericLength)
a060682 = genericLength . nub . a193829_row
-- Reinhard Zumkeller, Jun 25 2015

Mathematica

a[n_ ] := Length[Union[Drop[d=Divisors[n], 1]-Drop[d, -1]]]

PARI

a(n) = my(d=divisors(n)); #vecsort(vector(#d-1, k, d[k+1] - d[k]),,8); \\ Michel Marcus, Jul 04 2017

Extensions

Edited by Dean Hickerson, Jan 22 2002