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

2, 13, 17, 29, 37, 53, 73, 89, 97, 101, 113, 173, 181, 193, 197, 233, 241, 257, 277, 293, 313, 337, 349, 353, 373, 409, 421, 433, 449, 457, 521, 541, 569, 577, 593, 613, 641, 661, 673, 677, 709, 733, 757, 761, 809, 821, 853, 881, 929, 1021, 1033, 1049, 1069

Offset: 1

Author

Michel Lagneau, Mar 06 2010

Keywords

• nonn

Comments

One contribution to the set of solutions is from (x,y) where x and y are both prime, see A045637.

Another set of solutions is contributed if (x,y) are both in A026424.

Examples

			2 is in the sequence because 2 = 1 + 1 and L(1)*L(1)= (1) *(1) = 1.
13 is in the sequence because 13 = 2^2 + 3^2 and L(2)*L(3)= (-1)*(-1) = 1.
193 is in the sequence because 193 = 12^2 + 7^2 and L(12)*L(7)= (-1)*(-1) = 1.
		

Crossrefs

Cf. A002819, A002313, A174054.

Programs

Maple

isA174050 := proc(n)
        local x,y ;
        if not isprime(n) then
                return false;
        end if;
        for x from 1 do
                if x^2 > n then
                        return false;
                end if;
                if issqr(n-x^2) then
                        y := sqrt(n-x^2) ;
                        if A008836(x) * A008836(y) = 1 then
                                return true;
                        end if;
                end if;
        end do:
end proc:
for n from 1 to 1100 do
        if isA174050(n) then
                printf("%d,\n",n) ;
        end if;
end do: # R. J. Mathar, Jul 09 2012

Mathematica

lambdaQ[{x_, y_}] := LiouvilleLambda[x]*LiouvilleLambda[y] == 1; Select[ Prime /@ Range[200], Or @@ lambdaQ /@ PowersRepresentations[#, 2, 2] &] (* Jean-François Alcover, Jul 30 2013 *)