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%I #18 Jan 15 2018 15:33:50
%S 41,50,54,65,66,74,86,90,98,99,110,113,114,117,121,122,126,131,137,
%T 145,150,164,166,169,174,178,179,181,182,186,197,200,205,216,218,219,
%U 222,226,227,229,237,258,260,264,265,275,286,291,296,302
%N Numbers that are the sum of three squares (square 0 allowed) in exactly three ways.
%C These are the numbers for which A000164(a(n)) = 3.
%C 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 This sequence is a proper subsequence of A000378.
%H Alois P. Heinz, <a href="/A224443/b224443.txt">Table of n, a(n) for n = 1..1000</a>
%F 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 The sequence gives the increasingly ordered members of the set {m integer | A000164(m) = 3, m >= 0}.
%e 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 The representations [a,b,c] for n = 1, ..., 10, are:
%e n=1, 41: [0, 4, 5], [1, 2, 6], [3, 4, 4],
%e n=2, 50: [0, 1, 7], [0, 5, 5], [3, 4, 5],
%e n=3, 54: [1, 2, 7], [2, 5, 5], [3, 3, 6],
%e n=4, 65: [0, 1, 8], [0, 4, 7], [2, 5, 6],
%e n=5, 66: [1, 1, 8], [1, 4, 7], [4, 5, 5],
%e n=6, 74: [0, 5, 7], [1, 3, 8], [3, 4, 7],
%e n=7, 86: [1, 2, 9], [1, 6, 7], [5, 5, 6],
%e n=8, 90: [0, 3, 9], [1, 5, 8], [4, 5, 7],
%e n=9, 98: [0, 7, 7], [1, 4, 9], [3, 5, 8],
%e n=10, 99: [1, 7, 7], [3, 3, 9], [5, 5, 7].
%p b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i^2<n, 0,
%p `if`(b(n, i-1, t)>3, 4, min(4, b(n, i-1, t)+
%p `if`(i^2>n, 0, b(n-i^2, i, t-1))))))
%p end:
%p a:= proc(n) option remember; local k;
%p for k from 1 +`if`(n=1, 0, a(n-1))
%p while b(k, isqrt(k), 3)<>3 do od; k
%p end:
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Apr 09 2013
%t Select[ Range[0, 400], Length[ PowersRepresentations[#, 3, 2]] == 3 &] (* _Jean-François Alcover_, Apr 09 2013 *)
%Y Cf. A000164, A005875, A000378, A094942 (one way), A224442 (two ways).
%K nonn
%O 1,1
%A _Wolfdieter Lang_, Apr 08 2013
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