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

2,3

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.

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.

Example

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.

Mathematica

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

Prog

(Haskell)
import Data.List (nub, genericLength)
a060682 = genericLength . nub . a193829_row
-- Reinhard Zumkeller, Jun 25 2015
(PARI) a(n) = my(d=divisors(n)); #vecsort(vector(#d-1, k, d[k+1] - d[k]), , 8); \\ Michel Marcus, Jul 04 2017

Crossrefs

Cf. A000005, A060680, A060681, A060683.

Cf. A193829.

Sequence in context: A173442 A112309 A160006 * A352897 A280363 A217743

Adjacent sequences: A060679 A060680 A060681 * A060683 A060684 A060685

Keyword

nonn

Author

Labos Elemer, Apr 19 2001

Extensions

Edited by Dean Hickerson, Jan 22 2002

Status

approved