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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. 7
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..33.

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

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

Sequence in context: A317709 A034044 A047447 * A302494 A302534 A063451

Adjacent sequences:  A094736 A094737 A094738 * A094740 A094741 A094742

KEYWORD

nonn,fini,full

AUTHOR

T. D. Noe, May 24 2004

EXTENSIONS

Keyword full added by Wolfdieter Lang, Aug 27 2020

STATUS

approved

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Last modified January 22 00:08 EST 2022. Contains 350481 sequences. (Running on oeis4.)