A174054 Primes of the form x^2+y^2 such that L(x)*L(y) = -1, where L is the Liouville lambda-function A008836.
5, 41, 61, 109, 137, 149, 157, 229, 269, 281, 317, 389, 397, 401, 461, 509, 557, 601, 617, 653, 701, 769, 773, 797, 829, 857, 877, 937, 941, 977, 997, 1009, 1013, 1061, 1093, 1097, 1153, 1181, 1213, 1217, 1229, 1249, 1277, 1289, 1321, 1409, 1453, 1489
Offset: 1
Keywords
Examples
5 = 1 + 2^2 and L(1)*L(2)= (1) *(-1) = -1. 41 = 4^2 + 5^2 and L(4)*L(5)= (1)*(-1) = -1
Programs
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Maple
isA174054 := 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 2000 do if isA174054(n) then printf("%d,\n",n) ; end if; end do: # R. J. Mathar, Jul 09 2012
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Mathematica
lambdaQ[{x_, y_}] := LiouvilleLambda[x]*LiouvilleLambda[y] == -1; Select[ Prime /@ Range[300], Or @@ lambdaQ /@ PowersRepresentations[#, 2, 2] &] (* Jean-François Alcover, Jul 30 2013 *)