A077199 - OEIS

%I #17 May 24 2014 21:17:40

%S 2,3,2,2,2,5,3,2,2,3,2,2,2,3,2,3,2,3,2,2,2,7,3,2,5,3,2,2,2,3,2,2,2,3,

%T 2,2,2,3,2,2,2,5,3,2,2,5,6,3,2,3,2,3,2,3,2,2,2,3,2,2,5,3,2,2,2,3,2,2,

%U 2,3

%N Smallest k such that both k and k+n are squarefree.

%C If a(n) = 3 or 7 then a(n+1) = 2 or 6 respectively.

%C Conjecture: every term is < 10, i.e. for every n at least one of the numbers n+2, n+3, n+5, n+6 or n+7 is squarefree.

%C The conjecture is false.  Here are 9 counterexamples, each of which is less than 10000: 1857, 2522, 3570, 4470, 6169, 6645, 7981, 9553, 9745.  There are 16 counterexamples within the first 10000 squarefree numbers. - _Harvey P. Dale_, May 24 2014

%e a(12) = 2 as 2+12 = 14 is squarefree.

%t With[{sqfree=Select[Range[2,20],SquareFreeQ]},Flatten[ Table[ Select[ sqfree+ n, SquareFreeQ,1]-n,{n,70}]]] (* _Harvey P. Dale_, May 21 2014 *)

%o (PARI) a(n) = {k = 2; while(!issquarefree(k) || !issquarefree(k+n), k++); k;} \\ _Michel Marcus_, May 24 2014

%K nonn

%O 1,1

%A _Amarnath Murthy_, Nov 01 2002

%E Corrected and extended by _Harvey P. Dale_, May 21 2014