A020754 Increasing gaps between squarefree numbers (lower end).

1, 3, 7, 47, 241, 843, 22019, 217069, 1092746, 8870023, 221167421, 47255689914, 82462576219, 1043460553363, 79180770078547, 3215226335143217, 23742453640900971, 125781000834058567

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• hard
• nice

Comments

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

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

Examples

			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.
		

Links

• Tsz Ho Chan, New small gaps between squarefree numbers, arXiv:2110.09990 [math.NT], 2021. [Note: according to Pandey, Chan has discovered an error in this paper.]
• Louis Gordon, Mark F. Schilling, and Michael S. Waterman, An extreme value theory for long head runs, Probability Theory and Related Fields, Vol. 72 (1986), pp. 279-287.
• Angel Kumchev, Wade McCormick, Nathan McNew, Ariana Park, Russell Scherr, and Simon Ziehr, Explicit bounds for large gaps between squarefree and cubefree integers, arXiv preprint (2022). arXiv:2211.09975 [math.NT]
• Michael J. Mossinghoff, Tomás Oliveira e Silva, and Tim Trudgian, The distribution of k-free numbers, arXiv:1912.04972 [math.NT], 2019. See Table 3, p. 14.
• Mayank Pandey, Squarefree numbers in short intervals, arXiv preprint (2024). arXiv:2401.13981 [math.NT]

Crossrefs

Cf. A005117, A020753, A020755, A045882, A051681.

Programs

Mathematica

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 *)

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

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

Formula

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

Extensions

Thanks to Christian G. Bower for additional comments.

a(16)-a(18) from A045882 by Jens Kruse Andersen, May 01 2015