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%I #36 Jun 20 2022 05:43:17
%S 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 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 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 Gaps between squarefree numbers: a(n) = A005117(n+1) - A005117(n).
%C This sequence is unbounded, as a simple consequence of the Chinese remainder theorem. - _Thomas Ordowski_, Jul 22 2015
%C Conjecture: lim sup_{n->oo} a(n)/log(A005117(n)) = 1/2. - _Thomas Ordowski_, Jul 23 2015
%C 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 From _Amiram Eldar_, Mar 09 2021: (Start)
%C The asymptotic density of the occurrences of 1 in this sequence is density(A007674)/density(A005117) = A065474/A059956 = 0.530711... (A065469).
%C 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 Reinhard Zumkeller, <a href="/A076259/b076259.txt">Table of n, a(n) for n = 1..10000</a>
%F Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = Pi^2/6 (A013661). - _Amiram Eldar_, Oct 21 2020
%e 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 := proc(n) A005117(n+1)-A005117(n) ; end proc: # _R. J. Mathar_, Jan 09 2013
%t Select[Range[200], SquareFreeQ] // Differences (* _Jean-François Alcover_, Mar 10 2019 *)
%o (Haskell)
%o a076259 n = a076259_list !! (n-1)
%o a076259_list = zipWith (-) (tail a005117_list) a005117_list
%o -- _Reinhard Zumkeller_, Aug 03 2012
%o (PARI) t=1; for(n=2,1e3, if(issquarefree(n), print1(n-t", "); t=n)) \\ _Charles R Greathouse IV_, Jul 23 2015
%Y Cf. A005117, A013661, A020753 (records), A020754, A076260.
%Y Cf. A007674, A007675, A059956, A065469, A065474, A069977, A206256.
%K nonn,easy
%O 1,3
%A _Reinhard Zumkeller_, Oct 03 2002
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