OFFSET
1,2
COMMENTS
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.
From Wolfdieter Lang, Aug 27 2020: (Start)
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.
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.
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)
LINKS
Steven Arno, The imaginary quadratic fields of class number 4, Acta Arithmetica 60.4 (1992) 321 - 334.
Irving Kaplansky, Integers Uniquely Represented by Certain Ternary Forms, in "The Mathematics of Paul Erdős I", Ronald. L. Graham and Jaroslav Nešetřil (Eds.), Springer, 1997, pp. 86 - 94.
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No.8, October 1948, pp. 476-481.
EXAMPLE
The unique partitions of m*4^k into three squares are,
for m = 1:
1 = 1^2 + 0^2 + 0^2;
4 = 2^2 + 0^2 + 0^2;
16 = 4^2 + 0^2 + 0^2;
...
for m = 163:
163 = 9^2 + 9^2 + 1^2;
163*4 = 18^2 + 18^2 + 2^2;
163*16 = 36^2 + 36^2 + 4^2;
...
MATHEMATICA
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&]
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
T. D. Noe, May 24 2004
EXTENSIONS
Keyword full added by Wolfdieter Lang, Aug 27 2020
STATUS
approved