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A134406 Composite numbers of the form k^2 + 1.

%I A134406 #22 Oct 26 2021 16:50:23
%S A134406 10,26,50,65,82,122,145,170,226,290,325,362,442,485,530,626,730,785,
%T A134406 842,901,962,1025,1090,1157,1226,1370,1445,1522,1682,1765,1850,1937,
%U A134406 2026,2117,2210,2305,2402,2501,2602,2705,2810,3026,3250,3365,3482,3601,3722,3845
%N A134406 Composite numbers of the form k^2 + 1.
%C A134406 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(50) = [7;{2,7}], ..., where {} is denoted a period (we write {6} == {2,3}).
%H A134406 Robert Israel, <a href="/A134406/b134406.txt">Table of n, a(n) for n = 1..10000</a>
%F A134406 a(n) = 1 + A134407(n)^2. - _R. J. Mathar_, Oct 13 2019
%e A134406 10 is a term because 10 = 3^2 + 1 is composite,
%e A134406 26 is a term because 26 = 5^2 + 1 is composite,
%e A134406 50 is a term because 50 = 7^2 + 1 is composite.
%p A134406 ts_fn1:=proc(n) local i,tren,ans; ans:=[ ]: for i from 1 to n do tren := i^(2)+1: if (isprime(tren) = false) then ans:=[ op(ans), tren ]: fi od: RETURN(ans) end: ts_fn1(200);
%t A134406 Select[Range[70]^2+1,!PrimeQ[#]&] (* _Harvey P. Dale_, Aug 12 2012 *)
%o A134406 (PARI) for(n=3,99, if(!isprime(t=n^2+1), print1(t", "))) \\ _Charles R Greathouse IV_, Aug 29 2016
%o A134406 (Python)
%o A134406 from sympy import isprime
%o A134406 from itertools import count, takewhile
%o A134406 def aupto(limit):
%o A134406     form = takewhile(lambda x: x <= limit, (k**2+1 for k in count(1)))
%o A134406     return [number for number in form if not isprime(number)]
%o A134406 print(aupto(3845)) # _Michael S. Branicky_, Oct 26 2021
%Y A134406 Cf. A002496, A005574, A134407.
%Y A134406 Supersequence of A144255.
%K A134406 nonn
%O A134406 1,1
%A A134406 _Jani Melik_, Jan 18 2008