A337993 Numbers k such that L(k) < sigma(k) + k/Pi^2, where L(k) = floor(H(k) + exp(H(k)) * log(H(k))) and H(k) = Sum_{j=1..k} 1/j.

1, 2, 4, 6, 12, 24, 60, 120, 360, 2520, 5040

Offset: 1

Author

Peter Luschny, Oct 15 2020

Keywords

• nonn
• more

Comments

Conjecture: This sequence is finite.

References

• S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.

Links

• J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, arXiv:math/0008177 [math.NT], 2000-2001; Am. Math. Monthly 109 (#6, 2002), 534-543.
• Peter Luschny, Illustration: Bounds for A057641.
• S. Ramanujan, Highly Composite Numbers, Proc. London Math. Soc. 2, 14, 1915, page 15.

Crossrefs

Cf. A000203, A002201, A002388, A004490, A057640, A057641.

Cf. A001008, A002805.

Programs

Mathematica

A337993Q[n_] := With[{h = HarmonicNumber[n]}, Floor[h + Exp[h]*Log[h]] < DivisorSigma[1, n] + n/Pi^2];
Select[Range[5040], A337993Q] (* Paolo Xausa, Feb 01 2024 *)

Formula

k is a term of this sequence <==> A057640(k) < A000203(k) + k/A002388.