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 A000534 Numbers that are not the sum of 4 nonzero squares. 6

%I

%S 0,1,2,3,5,6,8,9,11,14,17,24,29,32,41,56,96,128,224,384,512,896,1536,

%T 2048,3584,6144,8192,14336,24576,32768,57344,98304,131072,229376,

%U 393216,524288,917504,1572864,2097152,3670016,6291456,8388608,14680064

%N Numbers that are not the sum of 4 nonzero squares.

%C For n > 15, a(n) = A006431(n-1). - _Thomas Ordowski_, Nov 18 2012

%D J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, p. 140.

%D L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 302.

%D E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, Theorem 3, pp. 74-75.

%H T. D. Noe, <a href="/A000534/b000534.txt">Table of n, a(n) for n = 1..100</a>

%H Pierre de la Harpe, <a href="http://images.math.cnrs.fr/Lagrange-et-la-variation-des.html">Lagrange et la variation des théorèmes</a>, Images des Mathématiques, CNRS, 2014.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,4).

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%F Consists of the numbers 0, 1, 3, 5, 9, 11, 17, 29, 41, 2*4^m, 6*4^m and 14*4^m (m >= 0). Compare A123069.

%F From 224 on, a(n) = 4*a(n-3).

%F Numbers n such that A025428(n) = 0.

%t q=22;lst={};Do[Do[Do[Do[z=a^2+b^2+c^2+d^2;If[z<=q^2+3,AppendTo[lst,z]],{d,q}],{c,q}],{b,q}],{a,q}];lst1=Union@lst lst={};Do[AppendTo[lst,n],{n,q^2+3}];lst2=lst Complement[lst2,lst1] (* _Vladimir Joseph Stephan Orlovsky_, Feb 07 2010 *)

%t Join[{0,1,2,3,5,6,8,9,11,14,17,24,29,32,41}, LinearRecurrence[{0, 0, 4}, {56, 96, 128}, 30]] (* _Jean-François Alcover_, Feb 09 2016 *)

%o (PARI) for(n=1,224,if(sum(a=1,n,sum(b=1,a,sum(c=1,b,sum(d=1,c,if(a^2+b^2+c^2+d^2-n,0,1)))))==0,print1(n,",")))

%o (PARI) {a(n)=if(n<1, 0, if(n<15, [1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41][n], [4, 7, 12][(n+1)%3+1]*2^((n+1)\3*2-7)))} /* _Michael Somos_, Apr 08 2006 */

%o (PARI) {a(n)=if(n<2, 0, if(n<16, [1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41][n-1], [4, 7, 12][n%3+1]*2^(n\3*2-7)))} /* _Michael Somos_, Apr 23 2006 */

%o (PARI) is(n)=my(k=if(n,n/4^valuation(n,4),2)); k==2 || k==6 || k==14 || setsearch([0, 1, 3, 5, 9, 11, 17, 29, 41], n) \\ _Charles R Greathouse IV_, Sep 03 2014

%Y Cf. A123069, A000414 (complement).

%K nonn,easy,nice,changed

%O 1,3

%A _N. J. A. Sloane_ and _J. H. Conway_

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Last modified March 21 01:18 EDT 2019. Contains 321356 sequences. (Running on oeis4.)