A218048 Numbers n such that n^2 + 1, (n+1)^2 + 1 and (n+2)^2 + 1 are divisible by a square.
5742, 10716, 11731, 22868, 23156, 25757, 29505, 37080, 62967, 65641, 71218, 71922, 73443, 82542, 84906, 87892, 100456, 100792, 104868, 121918, 128567, 136282, 140992, 142168, 160142, 169605, 184131, 191067, 194280, 226191, 230107, 255118, 256118, 261005
Offset: 1
Keywords
Examples
5742 is in the sequence because 5742^2+1, 5743^2+1 and 5744^2+1 are divisible by squares. 5742^2+1 = 5 * 17^2 * 22817; 5743^2+1 = 2 * 5^2 * 701 * 941; 5744^2+1 = 109^2 * 2777.
Programs
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Maple
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:
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Mathematica
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 Select[Range[261005], ! SquareFreeQ[#^2 + 1] && ! SquareFreeQ[(# + 1)^2 + 1] && ! SquareFreeQ[(# + 2)^2 + 1] &] (* T. D. Noe, Oct 22 2012 *) 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 *)