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%I #54 Apr 17 2024 04:44:28
%S 1,3,7,47,241,843,22019,217069,1092746,8870023,221167421,47255689914,
%T 82462576219,1043460553363,79180770078547,3215226335143217,
%U 23742453640900971,125781000834058567
%N Increasing gaps between squarefree numbers (lower end).
%C We only consider gaps that set new records. The first gap of size 12 occurs (at 221167421) before the first gap of size 11 (at 262315466) and so for n>10, the n-th term in this sequence does not correspond to the first gap of length n. See A020753. - _Nathan McNew_, Dec 02 2020
%C The length of these runs are significantly shorter than would be predicted by a naive random model (for such a model see, e.g., Gordon, Schilling, & Waterman). For example, with n = a(18) and p = 6/Pi^2 the expected largest run is about 77.9 with variance 6.7, while A020753(18) = 18 which is 23 standard deviations smaller. - _Charles R Greathouse IV_, Oct 29 2021
%H Tsz Ho Chan, <a href="https://arxiv.org/abs/2110.09990">New small gaps between squarefree numbers</a>, arXiv:2110.09990 [math.NT], 2021. [Note: according to Pandey, Chan has discovered an error in this paper.]
%H Louis Gordon, Mark F. Schilling, and Michael S. Waterman, <a href="http://blog.thegrandlocus.com/static/misc/Gordon_Schilling_Waterman_1986.pdf">An extreme value theory for long head runs</a>, Probability Theory and Related Fields, Vol. 72 (1986), pp. 279-287.
%H Angel Kumchev, Wade McCormick, Nathan McNew, Ariana Park, Russell Scherr, and Simon Ziehr, <a href="https://arxiv.org/abs/2211.09975">Explicit bounds for large gaps between squarefree and cubefree integers</a>, arXiv preprint (2022). arXiv:2211.09975 [math.NT]
%H Michael J. Mossinghoff, Tomás Oliveira e Silva, and Tim Trudgian, <a href="https://arxiv.org/abs/1912.04972">The distribution of k-free numbers</a>, arXiv:1912.04972 [math.NT], 2019. See Table 3, p. 14.
%H Mayank Pandey, <a href="https://arxiv.org/abs/2401.13981">Squarefree numbers in short intervals</a>, arXiv preprint (2024). arXiv:2401.13981 [math.NT]
%F a(n) = A020755(n) - A020753(n); also a(n) = A020754(n+[n>10]) - 1 at least for n < 19. - _M. F. Hasler_, Dec 28 2015
%e The first gap in A005117 occurs between 1 and 2 and has length 1. The next largest gap occurs between 3 and 5 and has length 2. The next largest gap is between 7 and 10 and has length 3. Etc.
%t Module[{nn=11*10^5,sf,df},sf=Select[Range[nn],SquareFreeQ];df=Differences[sf];DeleteDuplicates[ Thread[{Most[sf],df}],GreaterEqual[#1[[2]],#2[[2]]]&]][[;;,1]] (* _Harvey P. Dale_, May 24 2023 *)
%o (PARI) A020754(n)=for(k=L=1, 9e9, issquarefree(k)||next; k-L>=n&&return(L); L=k) \\ For illustrative purpose only, not useful for n>10. - _M. F. Hasler_, Dec 28 2015
%o (PARI) r=0; L=1; forsquarefree(n=2,10^8,t=n[1]-L; if(t>r,r=t; print1(L", ")); L=n[1]) \\ _Charles R Greathouse IV_, Oct 22 2021
%Y Cf. A005117, A020753, A020755, A045882, A051681.
%K nonn,hard,nice,changed
%O 1,2
%A _David W. Wilson_
%E Thanks to _Christian G. Bower_ for additional comments.
%E a(16)-a(18) from A045882 by _Jens Kruse Andersen_, May 01 2015
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