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.

1, 2, 5, 10, 13, 25, 37, 58, 85, 130

Offset: 1

Author

Eberhard R. Hilf, Dec 21 1999

Keywords

• nonn
• hard
• nice
• more

Comments

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.

It is not known whether 130 is the largest such number or if this is the start of an infinite series.

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

Examples

			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.
		

References

• E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 79 with p. 76.

Links

• H. P. Baltes and E. R. Hilf, Spectra of finite systems; BI-Verlag.
• H.-P. Baltes, Peter K. J. Draxl and Eberhard R. Hilf, Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik, Publications of the Small Systems Group Oldenburg, preprint, 1973.
• H.-P. Baltes, Peter K. J. Draxl and Eberhard R. Hilf, Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik, Journ. Reine Angewandte Mathematik, Vol. 268/269, 1974, 410-417.
• P. K. J. Draxl, Sommes de deux carrés qui ne sont pas sommes de trois carrés strictement positifs, Mémoires de la Société Mathématique de France, 37 (1974), p. 53-53.
• E. Grosswald, A. Calloway and J. Calloway, The representations of integers by three positive squares, 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]
• F. Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arith. 42 (1982) 11-20, p. 13.
• Eberhard R. Hilf, Publications
• Eberhard R. Hilf, Über den Oberflächenterm der Gesamtenergie der Atomkerne nach dem Fermigas-Modell, Diploma-thesis, Universität Frankfurt, Germany, 1963.
• E. R. Hilf and H. P. Baltes, 130 and the cube spectrum, unpublished
• E. R. Hilf, G. Suessmann, Surface Tension of nuclei according to the Fermi-gas-model, Physics Letters, Vol. 21, No. 6, p. 654-656, (1966).
• Index entries for sequences related to sums of squares

Programs

Mathematica

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 *)

Extensions

Grosswald et al. reference from N. J. A. Sloane, Jun 07 2000