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A173331 Second of two intermediate sequences for integral solution of A002144(n)=x^2+y^2.

%I A173331 #2 Mar 30 2012 18:51:05
%S A173331 2,2,13,2,31,4,2,55,8,81,4,91,99,105,133,10,6,2,10,181,183,227,8,237,
%T A173331 16,10,14,265,2,301,303,16,18,8,355,379,6,381,389,14,421,429,453,451,
%U A173331 487,20,531,543,20,24,585,24,18,16,637,631,655,12,651,675,22,731,26,741,757
%N A173331 Second of two intermediate sequences for integral solution of A002144(n)=x^2+y^2.
%C A173331 a(n) = A173330(n)*A010050(A005098(n)) mod A002144(n);
%C A173331 A002973(n) = MIN(a(n), A002144(n) - a(n)) / 2.
%D A173331 H. Davenport, The Higher Arithmetic (Cambridge University Press 7th ed., 1999), ch. V.3, p.122.
%F A173331 a(n) = ((2k)! / 2(k!))^2 mod p, where p = 4*k+1 = A002144(n).
%e A173331 n=7: A002144(7) = 53 = 4*13 + 1,
%e A173331 a(7) = A173330(7) * 26! mod 53 = 7*403291461126605635584000000 mod 53 = 2,
%e A173331 A002973(7) = MIN(2, 53 - 2) / 2 = 1;
%e A173331 n=8: A002144(8) = 61 = 4*15 + 1,
%e A173331 a(8) = A173330(8) * 30! mod 61 = 5*265252859812191058636308480000000 mod 61 = 55,
%e A173331 A002973(8) = MIN(55, 61 - 55) / 2 = 3.
%Y A173331 Cf. A123072, A000142.
%K A173331 nonn
%O A173331 1,1
%A A173331 _Reinhard Zumkeller_, Feb 16 2010