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A000378 Sums of three squares: numbers of the form x^2 + y^2 + z^2. 38
0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

An equivalent definition: numbers of the form x^2 + y^2 + z^2 with x,y,z >= 0.

Bourgain studies "the spatial distribution of the representation of a large integer as a sum of three squares, on the small and critical scale as well as their electrostatic energy. The main results announced give strong evidence to the thesis that the solutions behave randomly. This is in sharp contrast to what happens with sums of two or four or more square." Sums of two nonzero squares are A000404. - Jonathan Vos Post, Apr 03 2012.

The multiplicities for a(n) (if 0 <= x <= y <= z) are given as A000164(a(n)), n >= 1. Compare with A005875(a(n)) for integer x, y and z, and order taken into account. - Wolfdieter Lang, Apr 08 2013

a(n)^k is a member of this sequence for any k > 1. - Boris Putievskiy, May 05 2013

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 37.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 311.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Jean Bourgain, Peter Sarnak, Zeév Rudnick, Local statistics of lattice points on the sphere, arXiv:1204.0134 [math.NT], 2012-2015.

J. W. Cogdell, On sums of three squares, Journal de théorie des nombres de Bordeaux, 15 no. 1 (2003), p. 33-44.

L. E. Dickson, Integers represented by positive ternary quadratic forms, Bull. Amer. Math. Soc. 33 (1927), 63-70. See Theorem I.

P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 91. [?Broken link]

P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 91.

Eric W. Weisstein, Square Number

Index entries for sequences related to sums of squares

FORMULA

Legendre: a nonnegative integer is a sum of three squares iff it is not of the form 4^k m with m == 7 (mod 8).

n^(2k+1) is in the sequence iff n is in the sequence. - Ray Chandler, Feb 03 2009

Complement of A004215; complement of A000302(i)*A004771(j), i,j>=0. - Boris Putievskiy, May 05 2013

a(n) = 6n/5 + O(log n). - Charles R Greathouse IV, Mar 14 2014

EXAMPLE

a(1) = 0 = 0^2 + 0^2 + 0^2. A005875(0) = 1 = A000164(0).

a(9) = 9 = 0^2 + 0^2 + 3^2 =  1^2 +  2^2 + 2^2. A000164(9) = 2. A000164(9) = 30 = 2*3 + 8*3 (counting signs and order). - Wolfdieter Lang, Apr 08 2013

MAPLE

isA000378 := proc(n) # return true or false depending on n being in the list

    local x, y ;

    for x from 0 do

        if 3*x^2 > n then

            return false;

        end if;

        for y from x do

            if x^2+2*y^2 > n then

                break;

            else

                if issqr(n-x^2-y^2) then

                    return true;

                end if;

            end if;

        end do:

    end do:

end proc:

A000378 := proc(n) # generate A000378(n)

    option remember;

    local a;

    if n = 1 then

        0;

    else

        for a from procname(n-1)+1 do

            if isA000378(a) then

                return a;

            end if;

        end do:

    end if;

end proc:

seq(A000378(n), n=1..100) ; # R. J. Mathar, Sep 09 2015

MATHEMATICA

Take[Union[Flatten[Table[a^2+b^2+c^2, {a, 0, 16}, {b, 0, 16}, {c, 0, 16}]]], 234] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2010 *)

okQ[n_] := If[EvenQ[k = IntegerExponent[n, 2]], m = n/2^k; Mod[m, 8] != 7, True]; Select[Range[0, 100], okQ] (* Jean-François Alcover, Feb 08 2016, adapted from PARI *)

PROG

(PARI) isA000378(n)=my(k=valuation(n, 2)); if(k%2==0, n>>=k; n%8!=7, 1)

(PARI) list(lim)=my(v=List(), k, t); for(x=0, sqrtint(lim\=1), for(y=0, min(sqrtint(lim-x^2), x), k=x^2+y^2; for(z=0, min(sqrtint(lim-k), y), listput(v, k+z^2)))); Set(v) \\ Charles R Greathouse IV, Sep 14 2015

CROSSREFS

Union of {0}, A000290, A000404 and A000408. Complement of A004215.

Cf. A005875 (number of representations if x, y and z are integers).

Cf. A047449, A034043, A034044, A034045, A034046, A034047, A065883, A072400, A072401, A071374, A002828, A001481, A125084, A000164.

Sequence in context: A004777 A059561 A037474 * A022551 A172251 A286997

Adjacent sequences:  A000375 A000376 A000377 * A000379 A000380 A000381

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Ray Chandler, Sep 05 2004

STATUS

approved

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Last modified August 21 17:45 EDT 2017. Contains 290892 sequences.