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A174050 Primes of the form x^2 + y^2 such that L(x)* L(y) = 1, where L is the Liouville lambda-function A008836.

%I A174050 #8 Jul 30 2013 13:55:15
%S A174050 2,13,17,29,37,53,73,89,97,101,113,173,181,193,197,233,241,257,277,
%T A174050 293,313,337,349,353,373,409,421,433,449,457,521,541,569,577,593,613,
%U A174050 641,661,673,677,709,733,757,761,809,821,853,881,929,1021,1033,1049,1069
%N A174050 Primes of the form x^2 + y^2 such that L(x)* L(y) = 1, where L is the Liouville lambda-function A008836.
%C A174050 One contribution to the set of solutions is from (x,y) where x and y are both prime, see A045637.
%C A174050 Another set of solutions is contributed if (x,y) are both in A026424.
%e A174050 2 is in the sequence because 2 = 1 + 1 and L(1)*L(1)= (1) *(1) = 1.
%e A174050 13 is in the sequence because 13 = 2^2 + 3^2 and L(2)*L(3)= (-1)*(-1) = 1.
%e A174050 193 is in the sequence because 193 = 12^2 + 7^2 and L(12)*L(7)= (-1)*(-1) = 1.
%p A174050 isA174050 := proc(n)
%p A174050         local x,y ;
%p A174050         if not isprime(n) then
%p A174050                 return false;
%p A174050         end if;
%p A174050         for x from 1 do
%p A174050                 if x^2 > n then
%p A174050                         return false;
%p A174050                 end if;
%p A174050                 if issqr(n-x^2) then
%p A174050                         y := sqrt(n-x^2) ;
%p A174050                         if A008836(x) * A008836(y) = 1 then
%p A174050                                 return true;
%p A174050                         end if;
%p A174050                 end if;
%p A174050         end do:
%p A174050 end proc:
%p A174050 for n from 1 to 1100 do
%p A174050         if isA174050(n) then
%p A174050                 printf("%d,\n",n) ;
%p A174050         end if;
%p A174050 end do: # _R. J. Mathar_, Jul 09 2012
%t A174050 lambdaQ[{x_, y_}] := LiouvilleLambda[x]*LiouvilleLambda[y] == 1; Select[ Prime /@ Range[200], Or @@ lambdaQ /@ PowersRepresentations[#, 2, 2] &] (* _Jean-François Alcover_, Jul 30 2013 *)
%Y A174050 Cf. A002819, A002313, A174054.
%K A174050 nonn
%O A174050 1,1
%A A174050 _Michel Lagneau_, Mar 06 2010