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A169580 Squares of the form x^2+y^2+z^2 with x,y,z positive integers. 6
9, 36, 49, 81, 121, 144, 169, 196, 225, 289, 324, 361, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Integer solutions of a^2 = b^2 + c^2 + d^2, i.e., Pythagorean Quadruples. - Jon Perry, Oct 06 2012

Also null (or light-like, or isotropic) vectors in Minkowski 4-space. - Jon Perry, Oct 06 2012

REFERENCES

T. Nagell, Introduction to Number Theory, Wiley, 1951, p. 194.

LINKS

Robert Israel, Table of n, a(n) for n = 1..3140

O. Fraser and B. Gordon, On representing a square as the sum of three squares, Amer. Math. Monthly, 76 (1969), 922-923.

D. Rabahy, Spreadsheet revealing sequence in table of all a^2+b^2+c^2

David Rabahy, a^2+b^2+c^2 squares colored, zoomed way out to make "lines" apparent

EXAMPLE

9 = 1 + 4 + 4,

36 = 16 + 16 + 4,

49 = 36 + 9 + 4,

81 = 49 + 16 + 16,

so these are in the sequence.

16 cannot be written as the sum of 3 squares if zero is not allowed, therefore 16 is not in the sequence.

Also we can see that 49-36-9-4=0, so (7,6,3,2) is a null vector in the signatures (+,-,-,-) and (-,+,+,+). - Jon Perry, Oct 06 2012

MAPLE

M:= 10000: # to get all terms <= M

sort(convert(select(issqr, {seq(seq(seq(x^2 + y^2 + z^2,

  z=y..floor(sqrt(M-x^2-y^2))), y=x..floor(sqrt((M-x^2)/2))),

x=1..floor(sqrt(M/3)))}), list)); # Robert Israel, Jan 28 2016

MATHEMATICA

Select[Range[60]^2, Resolve@ Exists[{x, y, z}, Reduce[# == x^2 + y^2 + z^2, {x, y, z}, Integers], And[x > 0, y > 0, z > 0]] &] (* Michael De Vlieger, Jan 27 2016 *)

CROSSREFS

For the square roots see A005767. Cf. A000378, A000419.

Cf. A217554.

Sequence in context: A192610 A319958 A329808 * A258844 A294952 A068810

Adjacent sequences:  A169577 A169578 A169579 * A169581 A169582 A169583

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Mar 02 2010

STATUS

approved

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Last modified December 10 14:27 EST 2019. Contains 329896 sequences. (Running on oeis4.)