# square-free, cont.

N. J. A. Sloane njas at research.att.com
Sun Jan 14 21:19:59 CET 2001

```Subject: gaps between square-free numbers (more)

Someone said the description was not clear, so let me add
a few more words:

there's an old sequence of David Wilson's that needs a few more
terms - any volunteers?

The square-free numbers are sequence A005117:

%I A005117 M0617
%S A005117 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,30,31,33,34,35,37,38,39,
%T A005117 41,42,43,46,47,51,53,55,57,58,59,61,62,65,66,67,69,70,71,73,74,77,78,79,
%U A005117 82,83,85,86,87,89,91,93,94,95,97,101,102,103,105,106,107,109,110,111,113
%N A005117 Square-free numbers.
...

The zeroth gap occurs between 1 and 2 and has length 0.
The first gap occurs between 3 and 5 and has
length 1. The next gap is between 7 and 10 and has length 2. Etc.
We are only interested in gaps that set new records.

The sequence that needs extending is:

%I A020753
%S A020753 0,1,2,3,4,5,6,7,8,9,10,12
%N A020753 Sizes of successive increasing gaps between square-free numbers.
----------
%K A020753 nonn,more
%O A020753 1,3
%A A020753 dww
%Y A020753 Cf. A020754, A020755.

The two related sequences are

%I A020754
%S A020754 1,3,7,47,241,843,22019,217069,1092746,8870023,221167421
%N A020754 Increasing gaps between square-free numbers (lower end).
%K A020754 nonn
%O A020754 1,2
%A A020754 dww
%Y A020754 Cf. A020753, A020755.

%I A020755
%S A020755 2,5,10,51,246,849,22026,217077,1092755,8870033,221167433
%N A020755 Increasing gaps between square-free numbers (upper end).
%K A020755 nonn
%O A020755 1,1
%A A020755 dww
%Y A020755 Cf. A020754, A020753.

NJAS

```