A094739 - OEIS

%I #59 Aug 09 2022 14:07:53

%S 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 115,133,142,163,190,235,253,403,427

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

%C 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 From _Wolfdieter Lang_, Aug 27 2020: (Start)

%C 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 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 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 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 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 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 The unique partitions of m*4^k into three squares are,

%e for m = 1:

%e 1 = 1^2 + 0^2 + 0^2;

%e 4 = 2^2 + 0^2 + 0^2;

%e 16 = 4^2 + 0^2 + 0^2;

%e ...

%e for m = 163:

%e 163 = 9^2 + 9^2 + 1^2;

%e 163*4 = 18^2 + 18^2 + 2^2;

%e 163*16 = 36^2 + 36^2 + 4^2;

%e ...

%t 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 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 nonn,fini,full

%O 1,2

%A _T. D. Noe_, May 24 2004

%E Keyword full added by _Wolfdieter Lang_, Aug 27 2020