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%I #62 Apr 18 2024 11:38:41
%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 Brennan Benfield and Oliver Lippard, <a href="https://arxiv.org/abs/2404.08193">Integers that are not the sum of positive powers</a>, arXiv:2404.08193 [math.NT], 2024. See p. 2.
%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.
%F G.f.: x^2*(36*x^16 + 32*x^15 + 60*x^14 + 55*x^13 + 36*x^12 + 27*x^11 + 20*x^10 + 19*x^9 + 18*x^8 + 13*x^7 + 11*x^6 + 4*x^5 + 2*x^4 - x^3 - 3*x^2 - 2*x - 1)/(4*x^3 - 1). - _Chai Wah Wu_, Jul 09 2022
%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<2, 0, 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
%o (Python)
%o from itertools import count, islice
%o def A000534_gen(startvalue=0): # generator of terms >= startvalue
%o return filter(lambda n:n in {0, 1, 3, 5, 9, 11, 17, 29, 41} or n>>((~n&n-1).bit_length()&-2) in {2,6,14},count(max(startvalue,0)))
%o A000534_list = list(islice(A000534_gen(),30)) # _Chai Wah Wu_, Jul 09 2022
%Y Cf. A123069, A000414 (complement).
%K nonn,easy,nice
%O 1,3
%A _N. J. A. Sloane_ and _J. H. Conway_
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