A020753 - OEIS

%I #36 Mar 12 2022 22:42:47

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

%N Sizes of successive increasing gaps between squarefree numbers.

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

%C 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

%C Conjecture: a(n) ~ log(A020754(n))/2. - _Thomas Ordowski_, Jul 23 2015

%F a(n) = A020755(n) - A020754(n). - _M. F. Hasler_, Dec 28 2015

%e 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.

%p 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

%t Union[Differences[Select[Range[10000000], SquareFreeQ]]] (* _Harvey P. Dale_, May 04 2011 *)

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

%K nonn,hard,more,nice

%O 1,2

%A _David W. Wilson_

%E Thanks to _Christian G. Bower_ for additional comments.

%E More terms computed (using data from A020754) by _M. F. Hasler_, Dec 28 2015