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A218048 Numbers n such that n^2 + 1, (n+1)^2 + 1 and (n+2)^2 + 1 are divisible by a square. 1

%I #11 Aug 05 2019 11:26:59

%S 5742,10716,11731,22868,23156,25757,29505,37080,62967,65641,71218,

%T 71922,73443,82542,84906,87892,100456,100792,104868,121918,128567,

%U 136282,140992,142168,160142,169605,184131,191067,194280,226191,230107,255118,256118,261005

%N Numbers n such that n^2 + 1, (n+1)^2 + 1 and (n+2)^2 + 1 are divisible by a square.

%e 5742 is in the sequence because 5742^2+1, 5743^2+1 and 5744^2+1 are divisible by squares.

%e 5742^2+1 = 5 * 17^2 * 22817;

%e 5743^2+1 = 2 * 5^2 * 701 * 941;

%e 5744^2+1 = 109^2 * 2777.

%p with(numtheory):for n from 1 to 300000 do :x:=n^2+1:y:=(n+1)^2+1:z:= (n+2)^2+1:if issqrfree(x)=false and issqrfree(y)=false and issqrfree(z)=false then printf(`%d, `,n):else fi:od:

%t f[n_] := Union[ Transpose[ FactorInteger[n^2+1]] [[2]]] [[ -1]]; lst={};a = 0; b = 1; Do[c = f[n]; If[a> 1 && b > 1 && c > 1, AppendTo[lst,n-2]]; a = b; b = c, {n, 3, 5*10^5}]; lst

%t Select[Range[261005], ! SquareFreeQ[#^2 + 1] && ! SquareFreeQ[(# + 1)^2 + 1] && ! SquareFreeQ[(# + 2)^2 + 1] &] (* _T. D. Noe_, Oct 22 2012 *)

%t SequencePosition[Table[If[SquareFreeQ[n^2+1],0,1],{n,27*10^4}],{1,1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Aug 05 2019 *)

%Y Cf. A002522, A217798.

%K nonn

%O 1,1

%A _Michel Lagneau_, Oct 19 2012

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Last modified March 28 14:33 EDT 2024. Contains 371254 sequences. (Running on oeis4.)