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URL: https://oeis.org/A134420

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A134420
Composite squarefree numbers of the form k^2 + 1.
2
10, 26, 65, 82, 122, 145, 170, 226, 290, 362, 442, 485, 530, 626, 730, 785, 842, 901, 962, 1090, 1157, 1226, 1370, 1522, 1765, 1937, 2026, 2117, 2210, 2305, 2402, 2501, 2602, 2705, 2810, 3026, 3365, 3482, 3601, 3722, 3845, 3970, 4097, 4226, 4490, 4762
OFFSET
1,1
COMMENTS
Square roots of these numbers are quadratic irrationals and corresponding chain fraction representations are periodic: sqrt(10) = [3;{2,3}], sqrt(26) = [5;{2,5}], sqrt(65) = [8;{2,8}], ..., where {} is denoted a period (we write {6} == {2,3}).
LINKS
FORMULA
a(n) = A002522(A134427(n)). - Amiram Eldar, Feb 22 2021
EXAMPLE
a(1)=10 because 10 = 3^2 + 1 is squarefree.
a(2)=26 because 26 = 5^2 + 1 is squarefree.
a(3)=65 because 65 = 8^2 + 1 is squarefree.
MAPLE
ts_fn3:=proc(n) local i, tren, ans; ans:=[ ]: for i from 1 to n do tren := i^(2)+1: if (isprime(tren) = false and numtheory[mobius] (tren) <> 0 ) then ans:=[ op(ans), tren ]: fi od: RETURN(ans) end: ts_fn3(200);
MATHEMATICA
Select[Range[70]^2+1, CompositeQ[#] && SquareFreeQ[#] &] (* Amiram Eldar, Feb 22 2021 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Jani Melik, Jan 18 2008
EXTENSIONS
Definition corrected by T. D. Noe, Sep 16 2008
STATUS
approved