%I A077233
%S A077233 1,1,1,2,3,1,1,3,2,5,4,1,1,4,39,2,12,42,5,1,1,5,24,13,2,273,3,4,6,1,1,
%T A077233 6,4,3,5,2,531,30,24,3588,7,1,1,7,90,25,66,12,2,20,13,69,4,3805,8,1,1,
%U A077233 8,5967,4,936,30,413,2,125,5,3,6630,40,6,9,1,1,9,6,41,1122,3,21,53,2,165,
               120,1260,221064,4,5,569,10,1,1,10,22419
%N A077233 a(n) is smallest natural number satisfying Pell equation b^2- d(n)*a^2= 
               +1 or = -1, with d(n)=A000037(n) (non-square). Corresponding smallest 
               b(n)=A077232(n).
%C A077233 If d(n)=A000037(n) is from A003654 (that is if the regular continued 
               fraction for sqrt(d(n)) has odd (primitive) period length) then the 
               -1 option applies. For such d(n) the minimal b(n) and a(n) numbers 
               for the +1 option are 2*b(n)^2 + 1 and 2*b(n)*a(n), respectively 
               (see Perron I, pp. 94,p5).
%C A077233 For general integer solutions see A077232 comments.
%C A077233 If the trivial solution x=1, y=0 is included, the sequence becomes A006703. 
               - T. D. Noe (noe(AT)sspectra.com), May 17 2007
%D A077233 T. Nagell, "Introduction to Number Theory", Chelsea Pub., New York, 1964, 
               table p. 301.
%D A077233 O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 
               (Sec. 26, p. 91 with explanation on pp. 94,95).
%F A077233 a(n)=sqrt((A077232(n)^2 - (-1)^(c(n)))/A000037(n)) with c(n)=1 if A000037(n)=A003654(k) 
               for some k>=1 else c(n)=0.
%e A077233 d=10=A000037(7)=A003654(3), therefore a(7)=1 and b(7)=A077232(7)=3 give 
               3^2=10*1^2 -1 and 2*b(7)^2+1=19 and 2*b(7)*a(7)=2*3*1=6 satisfy 19^2 
               - 10*6^2 = +1.
%e A077233 d=11=A000037(8) is not in A003654, therefore there is no (nontrivial) 
               solution of the b^2 - d*a^2 = -1 Pell equation and a(8)=3 and b(8)=A077232(8)=10 
               satisfy 10^2 - 11*3^2 = +1. See A077232 for further examples.
%Y A077233 Cf. A033317, A003814.
%Y A077233 Sequence in context: A046226 A054722 A067627 this_sequence A123185 A133569 
               A141071
%Y A077233 Adjacent sequences: A077230 A077231 A077232 this_sequence A077234 A077235 
               A077236
%K A077233 nonn,nice
%O A077233 1,4
%A A077233 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 08 
               2002