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A051952
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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.
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6
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 79 with p. 76.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,hard,nice,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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