A285022 Numbers n such that A002088(n) < 3n^2/Pi^2.

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

Offset: 1

Author

Amiram Eldar, Apr 08 2017

Keywords

• nonn

Comments

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.

Examples

			A002088(820) = 204376, 3*820^2/(Pi^2) = 204385.091643... > 204376, thus 820 is in this sequence.
		

References

• 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.

Links

• Amiram Eldar and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 97 terms from Amiram Eldar)
• Paul Erdős and Harold N. Shapiro, On the Changes of Sign of a Certain Error Function, Canadian Journal of Mathematics, Vol. 3 (1951), pp. 375-385.
• R. A. MacLeod, The Minimum of Phi(x)/x^2, Journal of the London Mathematical Society, Vol. 1, No. 1 (1967), pp. 652-660.
• James Joseph Sylvester, Note sur le théoreme de Legendre citée dans une note insérée dans les Comptes rendus, Comptes rendus hebdomadaires des seances de l'Academie des sciences, Vol. 46 (1883), pp. 463-465.
• James Joseph Sylvester, On the Number of Fractions Contained in any "Farey series" of which the Limiting Number is Given, Philosophical Magazine, Series 5, Vol. 15, No. 94 (1883), pp. 251-257.

Crossrefs

Cf. A000010, A002088.

Programs

Maple

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

Mathematica

s = 0; k = 1; lst = {}; While[k < 50001, s = s + EulerPhi@k; If[s*Pi^2 < 3 k^2, AppendTo[lst, k]]; k++]; lst