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

1, 2, 3, 5, 6, 10, 11, 13, 14, 19, 21, 22, 30, 35, 37, 42, 43, 46, 58, 67, 70, 78, 91, 93, 115, 133, 142, 163, 190, 235, 253, 403, 427

Offset: 1

Author

T. D. Noe, May 24 2004

Keywords

• nonn
• fini
• full

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)

Examples

			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;
...
		

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.

Crossrefs

Cf. A005875 (number of ways of writing n as the sum of three squares), A094740 (n having a unique partition into three positive squares).

Programs

Mathematica

lim=100; nLst=Table[0, {lim^2}]; Do[n=a^2+b^2+c^2; If[n>0 && n0&]

Extensions

Keyword full added by Wolfdieter Lang, Aug 27 2020