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A224443 Numbers that are the sum of three squares (square 0 allowed) in exactly three ways.

%I A224443 #18 Jan 15 2018 15:33:50
%S A224443 41,50,54,65,66,74,86,90,98,99,110,113,114,117,121,122,126,131,137,
%T A224443 145,150,164,166,169,174,178,179,181,182,186,197,200,205,216,218,219,
%U A224443 222,226,227,229,237,258,260,264,265,275,286,291,296,302
%N A224443 Numbers that are the sum of three squares (square 0 allowed) in exactly three ways.
%C A224443 These are the numbers for which A000164(a(n)) = 3.
%C A224443 a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly three ways, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity of a(n) with order and signs taken into account is A005875(a(n)).
%C A224443 This sequence is a proper subsequence of A000378.
%H A224443 Alois P. Heinz, <a href="/A224443/b224443.txt">Table of n, a(n) for n = 1..1000</a>
%F A224443 This sequence gives the increasingly ordered numbers of the set {m integer | m = a^2 + b^2 + c^2, a, b and c integers with 0 <= a <= b <= c, and m has exactly three such representations}.
%F A224443 The sequence gives the increasingly ordered members of the set {m integer | A000164(m) = 3, m >= 0}.
%e A224443 a(1) = 41  = 0^2 + 4^2 + 5^2  = 1^2 + 2^2 + 6^2 = 3^3 + 4^2 + 4^2, and 41 is the first number m with A000164(m) = 3.
%e A224443 The representations [a,b,c] for n = 1, ..., 10, are:
%e A224443 n=1,  41: [0, 4, 5], [1, 2, 6], [3, 4, 4],
%e A224443 n=2,  50: [0, 1, 7], [0, 5, 5], [3, 4, 5],
%e A224443 n=3,  54: [1, 2, 7], [2, 5, 5], [3, 3, 6],
%e A224443 n=4,  65: [0, 1, 8], [0, 4, 7], [2, 5, 6],
%e A224443 n=5,  66: [1, 1, 8], [1, 4, 7], [4, 5, 5],
%e A224443 n=6,  74: [0, 5, 7], [1, 3, 8], [3, 4, 7],
%e A224443 n=7,  86: [1, 2, 9], [1, 6, 7], [5, 5, 6],
%e A224443 n=8,  90: [0, 3, 9], [1, 5, 8], [4, 5, 7],
%e A224443 n=9,  98: [0, 7, 7], [1, 4, 9], [3, 5, 8],
%e A224443 n=10, 99: [1, 7, 7], [3, 3, 9], [5, 5, 7].
%p A224443 b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i^2<n, 0,
%p A224443       `if`(b(n, i-1, t)>3, 4, min(4, b(n, i-1, t)+
%p A224443       `if`(i^2>n, 0, b(n-i^2, i, t-1))))))
%p A224443     end:
%p A224443 a:= proc(n) option remember; local k;
%p A224443       for k from 1 +`if`(n=1, 0, a(n-1))
%p A224443       while b(k, isqrt(k), 3)<>3 do od; k
%p A224443     end:
%p A224443 seq(a(n), n=1..100);  # _Alois P. Heinz_, Apr 09 2013
%t A224443 Select[ Range[0, 400], Length[ PowersRepresentations[#, 3, 2]] == 3 &] (* _Jean-François Alcover_, Apr 09 2013 *)
%Y A224443 Cf. A000164, A005875, A000378, A094942 (one way), A224442 (two ways).
%K A224443 nonn
%O A224443 1,1
%A A224443 _Wolfdieter Lang_, Apr 08 2013