This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A051952 #40 Aug 05 2021 14:42:34
%S A051952 1,2,5,10,13,25,37,58,85,130
%N A051952 Numbers that are not a sum of 3 positive squares nor are of the form 4^a*(8b+7) and which are not multiples of 4.
%C A051952 The asymptotic eigenvalue spectrum of the Schroedinger equation for a free particle in a box in three dimensions is known only (that is: average level density and average degeneracy) if the a(n) are finite series.
%C A051952 It is not known whether 130 is the largest such number or if this is the start of an infinite series.
%C A051952 See Theorem 6 of Grosswald's book, p. 79: every positive integer n not of the form 4^a*(8*m+7), with a and m nonnegative integers [i.e., n is a sum of three squares, proved by Legendre (1798)] and not of the form 4^b*a(n), with b a nonnegative integer, n=1,...,10, and possibly one a(11) > 5*10^10, is a sum of three non-vanishing squares. See A004214 and A000408. In the F. Halter-Koch reference, p. 13, one finds a Korollar 1 (b) stating for positive integers n, not 0, 4, 7 modulo 8: n is not the sum of three positive coprime squares if and only if n = a(n), n=1,...,10, and possibly one more number a(11) >= 5*10^10. See A223731. - _Wolfdieter Lang_, Apr 04 2013
%D A051952 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 79 with p. 76.
%H A051952 H. P. Baltes and E. R. Hilf, <a href="http://www.physik.uni-oldenburg.de/Docs/THEO3/publications/metadocs/ebs.spectra.finite.systems.prep.html">Spectra of finite systems</a>; BI-Verlag.
%H A051952 H.-P. Baltes, Peter K. J. Draxl and Eberhard R. Hilf, <a href="http://smallsystems.isn-oldenburg.de/publications/metadocs/ebs.quadratsummen.html">Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik</a>, Publications of the Small Systems Group Oldenburg, preprint, 1973.
%H A051952 H.-P. Baltes, Peter K. J. Draxl and Eberhard R. Hilf, <a href="https://doi.org/10.1515/crll.1974.268-269.410">Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik</a>, Journ. Reine Angewandte Mathematik, Vol. 268/269, 1974, 410-417.
%H A051952 P. K. J. Draxl, <a href="http://www.numdam.org/item?id=MSMF_1974__37__53_0">Sommes de deux carrés qui ne sont pas sommes de trois carrés strictement positifs</a>, Mémoires de la Société Mathématique de France, 37 (1974), p. 53-53.
%H A051952 E. Grosswald, A. Calloway and J. Calloway, <a href="https://doi.org/10.1090/S0002-9939-1959-0104623-5">The representations of integers by three positive squares</a>, Proc. Amer. Math. Soc. 10 (1959), 451-455. [Math. Rev. 21 #3376; E24-73 in Leveque's Reviews in Number Theory, Vol. 2, p. 290]
%H A051952 F. Halter-Koch, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4212.pdf">Darstellung natürlicher Zahlen als Summe von Quadraten</a>, Acta Arith. 42 (1982) 11-20, p. 13.
%H A051952 Eberhard R. Hilf, <a href="http://www.physik.uni-oldenburg.de/Docs/THEO3/publications/">Publications</a>
%H A051952 Eberhard R. Hilf, <a href="http://publikationen.ub.uni-frankfurt.de/frontdoor/index/index/docId/7612">Über den Oberflächenterm der Gesamtenergie der Atomkerne nach dem Fermigas-Modell</a>, Diploma-thesis, Universität Frankfurt, Germany, 1963.
%H A051952 E. R. Hilf and H. P. Baltes, <a href="http://www.physik.uni-oldenburg.de/Docs/THEO3/publications/metadocs/ebs.130.and.cube.spectrum.html">130 and the cube spectrum, unpublished</a>
%H A051952 E. R. Hilf, G. Suessmann, <a href="https://doi.org/10.1016/0031-9163(66)90113-2">Surface Tension of nuclei according to the Fermi-gas-model</a>, Physics Letters, Vol. 21, No. 6, p. 654-656, (1966).
%H A051952 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%e A051952 Consider a(3)=5: 1^2 +1^2 +1^2=3, too low; 1^2+1^2+2^2=6, too high. 4^1=4 too low; 4^2=16 too high; (8*0+7)=7 too low, (8*1+7)= 15 too high; thus 5 is a member of this sequence.
%t A051952 nmax = 1000; amax = Ceiling[Log[nmax/7]/Log[4]]; notThreeSquaresQ[n_] := Reduce[0 < a <= b <= c && n == a^2 + b^2 + c^2, {a, b, c}, Integers] === False; notOfTheFormQ[n_, a_] := Reduce[n == 4^a*(8*b+7), b, Integers] === False; notOfTheFormQ[n_] := And @@ (notOfTheFormQ[n, #] & ) /@ Range[0, amax]; Select[Range[nmax], !Mod[#, 4] == 0 && notThreeSquaresQ[#] && notOfTheFormQ[#] & ](* _Jean-François Alcover_, Jun 12 2012 *)
%K A051952 nonn,hard,nice,more
%O A051952 1,2
%A A051952 _Eberhard R. Hilf_, Dec 21 1999
%E A051952 Grosswald et al. reference from _N. J. A. Sloane_, Jun 07 2000