A020753 Sizes of successive increasing gaps between squarefree numbers.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• hard
• more
• nice

Comments

The indices of the records in A076259 are 1, 3, 6, 31, 150, 515, 13391, 131964, 664313, ... - R. J. Mathar, Jun 25 2010

Applying the test to squarefree numbers up to 10 million only produces the first ten terms of the sequence. - Harvey P. Dale, May 04 2011

Conjecture: a(n) ~ log(A020754(n))/2. - Thomas Ordowski, Jul 23 2015

Examples

			The first gap in A005117 occurs between 1 and 2 and has length 1. The next larger gap occurs between 3 and 5 and has length 2. The next larger gap is between 7 and 10 and has length 3. Etc. We are only interested in gaps that set new records.
		

Crossrefs

Cf. A005117, A020754, A020755, A045882, A051681, A076259.

Programs

Maple

a := 1 ; for n from 2 do if A076259(n) > a then print(n,A076259(n)) ; a := A076259(n) ; end if; end do: # R. J. Mathar, Jun 25 2010

Mathematica

Union[Differences[Select[Range[10000000], SquareFreeQ]]] (* Harvey P. Dale, May 04 2011 *)

Formula

a(n) = A020755(n) - A020754(n). - M. F. Hasler, Dec 28 2015

Extensions

Thanks to Christian G. Bower for additional comments.

More terms computed (using data from A020754) by M. F. Hasler, Dec 28 2015