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

%I A076259 #51 Dec 19 2024 21:15:50
%S A076259 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,
%T A076259 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,
%U A076259 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
%N A076259 Gaps between squarefree numbers: a(n) = A005117(n+1) - A005117(n).
%C A076259 This sequence is unbounded, as a simple consequence of the Chinese remainder theorem. - _Thomas Ordowski_, Jul 22 2015
%C A076259 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]
%C A076259 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
%C A076259 From _Amiram Eldar_, Mar 09 2021: (Start)
%C A076259 The asymptotic density of the occurrences of 1 in this sequence is density(A007674)/density(A005117) =  A065474/A059956 = 0.530711... (A065469).
%C A076259 The asymptotic density of the occurrences of 2 in this sequence is (density(A069977)-density(A007675))/density(A005117) =  (A065474-A206256)/A059956 = 0.324294...  (End)
%H A076259 Reinhard Zumkeller, <a href="/A076259/b076259.txt">Table of n, a(n) for n = 1..10000</a>
%H A076259 Mayank Pandey, <a href="https://arxiv.org/abs/2401.13981">Squarefree numbers in short intervals</a>, arXiv preprint, arXiv:2401.13981 [math.NT], 2024.
%F A076259 Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = Pi^2/6 (A013661). - _Amiram Eldar_, Oct 21 2020
%F A076259 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
%e A076259 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.
%p A076259 A076259 := proc(n) A005117(n+1)-A005117(n) ; end proc: # _R. J. Mathar_, Jan 09 2013
%t A076259 Select[Range[200], SquareFreeQ] // Differences (* _Jean-François Alcover_, Mar 10 2019 *)
%o A076259 (Haskell)
%o A076259 a076259 n = a076259_list !! (n-1)
%o A076259 a076259_list = zipWith (-) (tail a005117_list) a005117_list
%o A076259 -- _Reinhard Zumkeller_, Aug 03 2012
%o A076259 (PARI) t=1; for(n=2,1e3, if(issquarefree(n), print1(n-t", "); t=n)) \\ _Charles R Greathouse IV_, Jul 23 2015
%o A076259 (Python)
%o A076259 from math import isqrt
%o A076259 from sympy import mobius
%o A076259 def A076259(n):
%o A076259     def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A076259     m, k = n, f(n)
%o A076259     while m != k:
%o A076259         m, k = k, f(k)
%o A076259     r, k = n+1, f(n+1)+1
%o A076259     while r != k:
%o A076259         r, k = k, f(k)+1
%o A076259     return int(r-m) # _Chai Wah Wu_, Aug 15 2024
%Y A076259 Cf. A005117, A013661, A020753 (records), A020754, A076260.
%Y A076259 Cf. A007674, A007675, A059956, A065469, A065474, A069977, A206256.
%K A076259 nonn,easy
%O A076259 1,3
%A A076259 _Reinhard Zumkeller_, Oct 03 2002