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A094739 Numbers m such that 4^k*m, for integer k >= 0, are numbers having a unique partition into three squares.

%I A094739 #59 Aug 09 2022 14:07:53
%S A094739 1,2,3,5,6,10,11,13,14,19,21,22,30,35,37,42,43,46,58,67,70,78,91,93,
%T A094739 115,133,142,163,190,235,253,403,427
%N A094739 Numbers m such that 4^k*m, for integer k >= 0, are numbers having a unique partition into three squares.
%C A094739 Lehmer's paper has an erroneous version of this sequence. He omits 163 and includes 162 (which has 4 partitions) and 182 (which has 3 partitions). Lehmer conjectures that there are no more terms. Note that squares are allowed to be zero.
%C A094739 From _Wolfdieter Lang_, Aug 27 2020: (Start)
%C A094739 Another name is: Integers not divisible by 4 that are uniquely represented as x^2 + y^2 + z^2 with integers 0 <= x <= y <= z.
%C A094739 This sequence of 33 numbers is complete. See Arno, Theorem 8, p. 332, where 19 is missing, as observed by Kaplansky, Remark 2.1. (a) - (c), p. 87.
%C A094739 All positive integers represented uniquely as sum of three squares of nonnegative numbers, ignoring order and signs, are given by 4^k*a(n), for integer k >= 0 and n = 1 .. 33. See Arno, also p. 322, with some known results, and Kaplansky's Remark 2.1.(c). (End)
%H A094739 Steven Arno, <a href="https://doi.org/10.4064/aa-60-4-321-334">The imaginary quadratic fields of class number 4</a>, Acta Arithmetica 60.4 (1992) 321 - 334.
%H A094739 Irving Kaplansky, <a href="https://doi.org/10.1007/978-3-642-60408-9_6">Integers Uniquely Represented by Certain Ternary Forms</a>, in "The Mathematics of Paul Erdős I", Ronald. L. Graham and Jaroslav Nešetřil (Eds.), Springer, 1997, pp. 86 - 94.
%H A094739 D. H. Lehmer, <a href="http://www.jstor.org/stable/2305380">On the Partition of Numbers into Squares</a>, The American Mathematical Monthly, Vol. 55, No.8, October 1948, pp. 476-481.
%e A094739 The unique partitions of m*4^k into three squares are,
%e A094739 for m = 1:
%e A094739 1 = 1^2 + 0^2 + 0^2;
%e A094739 4 = 2^2 + 0^2 + 0^2;
%e A094739 16 = 4^2 + 0^2 + 0^2;
%e A094739 ...
%e A094739 for m = 163:
%e A094739 163 = 9^2 + 9^2 + 1^2;
%e A094739 163*4 = 18^2 + 18^2 + 2^2;
%e A094739 163*16 = 36^2 + 36^2 + 4^2;
%e A094739 ...
%t A094739 lim=100; nLst=Table[0, {lim^2}]; Do[n=a^2+b^2+c^2; If[n>0 && n<lim^2, nLst[[n]]++ ], {a, 0, lim}, {b, a, Sqrt[lim^2-a^2]}, {c, b, Sqrt[lim^2-a^2-b^2]}]; Select[Flatten[Position[nLst, 1]], Mod[ #, 4]>0&]
%Y A094739 Cf. A005875 (number of ways of writing n as the sum of three squares), A094740 (n having a unique partition into three positive squares).
%K A094739 nonn,fini,full
%O A094739 1,2
%A A094739 _T. D. Noe_, May 24 2004
%E A094739 Keyword full added by _Wolfdieter Lang_, Aug 27 2020