A094942 Numbers having a unique partition into three squares.

0, 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 16, 19, 20, 21, 22, 24, 30, 32, 35, 37, 40, 42, 43, 44, 46, 48, 52, 56, 58, 64, 67, 70, 76, 78, 80, 84, 88, 91, 93, 96, 115, 120, 128, 133, 140, 142, 148, 160, 163, 168, 172, 176, 184, 190, 192, 208, 224, 232, 235

Offset

1,3

Comments

Note that squares are allowed to be zero.

From Wolfdieter Lang, Apr 09 2013: (Start)

These are the numbers for which A000164(a(n)) = 1.

a(n) is the n-th largest number which has a representation as a sum of three squares (square 0 allowed), in exactly one way, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity with order and signs taken into account are A005875(a(n)).

These numbers are a proper subset of A000378.

(End)

Note that all these numbers are 4^k * A094739(n) for some k >= 0. - T. D. Noe, Nov 08 2013

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Formula

The sequence gives the increasingly ordered members of the set {m integer | A000164(m) = 1, m >= 0}.

Example

From Wolfdieter Lang, Apr 09 2013 (Start)
a(1) = 0 because 0 = 0^2 + 0^2 + 0^2 and 0 is the first number m with A000164(m)=1.
a(8) = 8 = 0^2 + 2^2 + 2^2, the 8th largest number m for which A000164(m) is 1.
(End)

Mathematica

lim=25; 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]}]; Flatten[Position[nLst, 1]]
Select[Range[0, 235], Length@PowersRepresentations[#, 3, 2] == 1 &] (* Ray Chandler, Oct 31 2019 *)

Crossrefs

Cf. A025321 (numbers having a unique partition into three positive squares), A094739 (primitive n having a unique partition into three squares).

Cf. A000164, A005875, A000378, A224442 (two ways), A224443 (three ways).

Sequence in context: A178596 A034047 A047424 * A243494 A302505 A102705

Adjacent sequences: A094939 A094940 A094941 * A094943 A094944 A094945

Keyword

nonn

Author

T. D. Noe, May 24 2004

Extensions

0 added by T. D. Noe, Apr 09 2013

Status

approved