A076259 Gaps between squarefree numbers: a(n) = A005117(n+1) - A005117(n).

1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 4, 2, 2, 2, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 4, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 3, 1, 4, 2, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 2, 1, 3, 2, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1

Offset: 1

Author

Reinhard Zumkeller, Oct 03 2002

Keywords

• nonn
• easy

Comments

This sequence is unbounded, as a simple consequence of the Chinese remainder theorem. - Thomas Ordowski, Jul 22 2015

Conjecture: lim sup_{n->oo} a(n)/log(A005117(n)) = 1/2. - Thomas Ordowski, Jul 23 2015 [Note: this conjecture is equivalent to lim sup a(n)/log n = 1/2. - Charles R Greathouse IV, Dec 05 2024]

a(n) = 1 infinitely often since the density of the squarefree numbers, 6/Pi^2, is greater than 1/2. In particular, at least 2 - Pi^2/6 = 35.5...% of the terms are 1. - Charles R Greathouse IV, Jul 23 2015

From Amiram Eldar, Mar 09 2021: (Start)

The asymptotic density of the occurrences of 1 in this sequence is density(A007674)/density(A005117) = A065474/A059956 = 0.530711... (A065469).

The asymptotic density of the occurrences of 2 in this sequence is (density(A069977)-density(A007675))/density(A005117) = (A065474-A206256)/A059956 = 0.324294... (End)

Examples

			As 24 = 3*2^3 and 25 = 5^2, the next squarefree number greater A005117(16) = 23 is A005117(17) = 26, therefore a(16) = 26-23 = 3.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Mayank Pandey, Squarefree numbers in short intervals, arXiv preprint, arXiv:2401.13981 [math.NT], 2024.

Crossrefs

Cf. A005117, A013661, A020753 (records), A020754, A076260.

Cf. A007674, A007675, A059956, A065469, A065474, A069977, A206256.

Programs

Haskell

a076259 n = a076259_list !! (n-1)
a076259_list = zipWith (-) (tail a005117_list) a005117_list
-- Reinhard Zumkeller, Aug 03 2012

Maple

A076259 := proc(n) A005117(n+1)-A005117(n) ; end proc: # R. J. Mathar, Jan 09 2013

Mathematica

Select[Range[200], SquareFreeQ] // Differences (* Jean-François Alcover, Mar 10 2019 *)

PARI

t=1; for(n=2,1e3, if(issquarefree(n), print1(n-t", "); t=n)) \\ Charles R Greathouse IV, Jul 23 2015

Python

from math import isqrt
from sympy import mobius
def A076259(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    r, k = n+1, f(n+1)+1
    while r != k:
        r, k = k, f(k)+1
    return int(r-m) # Chai Wah Wu, Aug 15 2024

Formula

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = Pi^2/6 (A013661). - Amiram Eldar, Oct 21 2020

a(n) < n^(1/5) for large enough n by a result of Pandey. (The constant Pi^2/6 can be absorbed by any eta > 0.) - Charles R Greathouse IV, Dec 04 2024