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A025321 Numbers that are the sum of 3 nonzero squares in exactly 1 way.

%I A025321 #37 Feb 16 2025 08:32:35
%S A025321 3,6,9,11,12,14,17,18,19,21,22,24,26,29,30,34,35,36,42,43,44,45,46,48,
%T A025321 49,50,53,56,61,65,67,68,70,72,73,76,78,82,84,88,91,93,96,97,104,106,
%U A025321 109,115,116,120,133,136,140,142,144,145,157,163,168,169,172,176,180,184,190
%N A025321 Numbers that are the sum of 3 nonzero squares in exactly 1 way.
%C A025321 It appears that all terms have the form 4^i A094740(j) for some i and j. - _T. D. Noe_, Jun 06 2008
%C A025321 This is true, because A025427(4*n) = A025427(n) for all n. - _Robert Israel_, Mar 09 2016
%H A025321 Donovan Johnson, <a href="/A025321/b025321.txt">Table of n, a(n) for n = 1..605</a> (terms < 10^8; first 417 terms from T. D. Noe)
%H A025321 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquareNumber.html">Square Number</a>.
%H A025321 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A025321 A243148(a(n),3) = 1. - _Alois P. Heinz_, Feb 25 2019
%t A025321 lim=20; nLst=Table[0, {lim^2}]; Do[n=a^2+b^2+c^2; If[n>0 && n<lim^2, nLst[[n]]++ ], {a, lim}, {b, a, Sqrt[lim^2-a^2]}, {c, b, Sqrt[lim^2-a^2-b^2]}]; Flatten[Position[nLst, 1]] (* _T. D. Noe_, Jun 06 2008 *)
%t A025321 b[n_, i_, k_, t_] := b[n, i, k, t] = If[n == 0, If[t == 0, 1, 0], If[i<1 || t<1, 0, b[n, i - 1, k, t] + If[i^2 > n, 0, b[n - i^2, i, k, t - 1]]]];
%t A025321 T[n_, k_] := b[n, Sqrt[n] // Floor, k, k];
%t A025321 Position[Table[T[n, 3], {n, 0, 200}], 1] - 1 // Flatten (* _Jean-François Alcover_, Nov 06 2020, after _Alois P. Heinz_ in A243148 *)
%o A025321 (PARI) is(n)=if(n<11, return(n>0 && n%3==0)); if(n%4==0, return(is(n/4))); my(w); for(i=sqrtint((n-1)\3)+1,sqrtint(n-2), my(t=n-i^2); for(j=sqrtint((t-1)\2)+1,min(sqrtint(t-1),i), if(issquare(t-j^2), w++>1 && return(0)))); w \\ _Charles R Greathouse IV_, Aug 05 2024
%Y A025321 Cf. A000408, A025427, A243148.
%K A025321 nonn
%O A025321 1,1
%A A025321 _David W. Wilson_