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A285022
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Numbers n such that A002088(n) < 3n^2/Pi^2.
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3
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820, 1276, 1926, 2080, 2640, 3160, 3186, 3250, 4446, 4720, 4930, 5370, 6006, 6546, 7386, 7450, 7476, 9066, 9276, 10626, 10836, 13146, 13300, 15640, 15666, 16056, 16060, 16446, 17020, 17466, 17550, 17766, 18040, 18910, 19176, 19230, 19416, 20736, 21000, 21246
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OFFSET
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1,1
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COMMENTS
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James Joseph Sylvester conjectured in 1883 that A002088(n) > 3n^2/Pi^2 for all n.
M. L. N. Sarma found the first counterexample, 820, in 1936.
Paul Erdős and Harold N. Shapiro proved in 1951 that A002088(n)- 3n^2/Pi^2 changes signs at infinitely many values of n, thus this sequence is infinite.
R. A. MacLeod proved in 1987 that A002088(n)/n^2 - 3/Pi^2 has a minimum at the second term, 1276.
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REFERENCES
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Sukumar Das Adhikari, The Average Behaviour of the Number of Solutions of a Diophantine Equation and an Averaging Technique, Number Theory: Diophantine, Computational, and Algebraic Aspects: Proceedings of the International Conference Held in Eger, Hungary, July 29-August 2, 1996. Walter de Gruyter, 1998.
Władysław Narkiewicz, Rational Number Theory in the 20th Century, Springer London, 2012, p. 215.
M. L. N. Sarma, On the Error Term in a Certain Sum, Proceedings of the Indian Academy of Sciences, Section A, Vol. 3, No. 1 (1936), pp. 338-338.
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LINKS
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EXAMPLE
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A002088(820) = 204376, 3*820^2/(Pi^2) = 204385.091643... > 204376, thus 820 is in this sequence.
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MAPLE
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F:= ListTools:-PartialSums(map(numtheory:-phi, [$1..30000])):
select(t -> is(F[t] < 3*t^2/Pi^2), [$1..30000]); # Robert Israel, Apr 21 2017
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MATHEMATICA
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s = 0; k = 1; lst = {}; While[k < 50001, s = s + EulerPhi@k; If[s*Pi^2 < 3 k^2, AppendTo[lst, k]]; k++]; lst
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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