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A224444 Multiplicities for representations of nonnegative numbers as primitive sums of three squares of integers (square 0 allowed).

%I A224444 #16 Apr 10 2013 11:58:39
%S A224444 0,1,1,1,0,1,1,0,0,1,1,1,0,1,1,0,0,2,1,1,0,1,1,0,0,1,2,1,0,2,1,0,0,2,
%T A224444 2,1,0,1,2,0,0,3,1,1,0,1,1,0,0,1,2,2,0,2,2,0,0,2,1,2,0,2,2,0,0,3,3,1,
%U A224444 0,2,1,0,0,2,3,1,0,2,1,0,0,2,2,2,0,2,3,0,0,4,2,1,0,1,2,0,0,2,2,2,0,4,2,0,0,2,2,2,0,2,3,0,0,3,3,1,0,2,2,0,0,2,3,2,0,3
%N A224444 Multiplicities for representations of nonnegative numbers as primitive sums of three squares of integers (square 0 allowed).
%C A224444 a(n) = 0, for n >= 1, if there is no representation of n as a sum of three squares (square 0 allowed) with no common factor > 1. a(0) = 0 because gcd(0,0,0) = 0 (not 1). a(n) = k >= 1 if n is representable as a primitive sum of three squares (square 0 allowed) in exactly k ways, if neither the order of the three terms nor the signs of the numbers to be squared are taken into account.
%C A224444 Compare with the multiplicities A000164.
%C A224444 The numbers for which a(n) is not 0 are given in A047449.
%F A224444 a(n) = k if n, for n >= 0, has exactly k representations n = a^2 + b^2 + c^2, with a, b and c integers, 0 <= a <= b < = c and gcd(a,b,c) = 1. If there is no such representation a(n) = 0.
%e A224444 a(0) = 0 because  0 = 0^2 + 0^2 + 0^2 is the only candidate for a representation but this is not a primitive sum because gcd(0,0,0) = 0, not 1.
%e A224444 a(2) = 1 because the only candidate for a representation of 2 is the triple [a,b,c] = [0,1,1] and this is primitive, because gcd(0,1,1) = 1.
%e A224444 a(9) = 1 because the two candidate triples are [0, 0, 3] and [1, 2, 2] but [0, 0, 3] is not primitive (gcd(0,0,3) =  3). A000164(9) = 2.
%e A224444 a(17) = 2 with the primitive [a,b,c] triples [0, 1, 4] and [2, 2, 3]. A000164(17) = 2 also.
%e A224444 a(41) = 3 = A000164(41) because the candidate triples [0, 4, 5], [1, 2, 6] and [3, 4, 4] are all primitive.
%t A224444 Table[ Count[ PowersRepresentations[n, 3, 2], pr_ /; GCD @@ pr == 1], {n, 0, 125}] (* _Jean-François Alcover_, Apr 09 2013 *)
%Y A224444 Cf. A047449, A000164.
%K A224444 nonn
%O A224444 0,18
%A A224444 _Wolfdieter Lang_, Apr 08 2013