A053797 Lengths of successive gaps between squarefree numbers.

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

Offset: 1

Author

N. J. A. Sloane, Apr 07 2000

Keywords

• nonn
• easy

Comments

From Gus Wiseman, Jun 11 2024: (Start)

Also the length of the n-th maximal run of nonsquarefree numbers. These runs begin:

4

8 9

12

16

18

20

24 25

27 28

32

36

40

44 45

48 49 50

(End)

Examples

			The first gap is at 4 and has length 1; the next starts at 8 and has length 2 (since neither 8 nor 9 are squarefree).
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..10000
• M. Filaseta and O. Trifonov, On Gaps between Squarefree Numbers. In Analytic Number Theory, Vol 85, 1990, Birkhäuser, Basel, pp. 235-253.
• E. Fogels, On the average values of arithmetic functions, Proc. Cambridge Philos. Soc. 1941, 37: 358-372.
• L. Marmet, First occurrences of squarefree gaps...
• L. Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012. - From _N. J. A. Sloane_, Jan 01 2013
• K. F. Roth, On the gaps between squarefree numbers, J. London Math. Soc. 1951 (2) 26:263-268.
• Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

Crossrefs

Gaps between terms of A005117.

For squarefree runs we have A120992, antiruns A373127 (firsts A373128).

For composite runs we have A176246 (rest of A046933), antiruns A373403.

For prime runs we have A251092 (rest of A175632), antiruns A027833.

Position of first appearance of n is A373199(n).

For antiruns instead of runs we have A373409.

A005117 lists the squarefree numbers, first differences A076259.

A013929 lists the nonsquarefree numbers, first differences A078147.

Cf. A007674, A020754, A045882, A053806, A061398, A061399.

Programs

Maple

SF:= select(numtheory:-issqrfree,[$1..1000]):
map(`-`,select(`>`,SF[2..-1]-SF[1..-2],1),1); # Robert Israel, Sep 22 2015

Mathematica

ReplaceAll[Differences[Select[Range@384, SquareFreeQ]] - 1, 0 -> Nothing] (* Michael De Vlieger, Sep 22 2015 *)

Extensions

Offset set to 1 by Peter Kagey, Sep 29 2015