A377570 a(n) = round((H(n)+e^H(n)*log(H(n))+sigma(n))/2).

1, 3, 5, 7, 8, 12, 12, 16, 17, 21, 19, 28, 23, 29, 30, 35, 30, 42, 34, 46, 43, 46, 41, 61, 48, 55, 55, 65, 53, 76, 57, 74, 68, 73, 71, 94, 69, 82, 81, 100, 77, 106, 81, 103, 101, 100, 89, 129, 97, 116, 107, 122, 102, 136, 114, 139, 121, 127, 114, 169, 118, 137

Offset

1,2

Comments

The idea of this sequence is finding the nearest integer to the arithmetic mean between the two sides of the inequality that is equivalent to the Riemann hypothesis. The inequality is sigma(n)<H(n)+e^H(n)*log(H(n)), where H(n) are the harmonic numbers.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

For n=6, sigma(6)=12 and H(6)=1/1+1/2+1/3+1/4+1/5+1/6=2.45 and (2.45+e^2.45*log(2.45)+12)/2=12.417... and round(12.417...)=12; hence a(6)=12.

Mathematica

f[x_] := x + Exp[x] * Log[x]; a[n_] := Round[(f[HarmonicNumber[n]] + DivisorSigma[1, n])/2]; Array[a, 100] (* Amiram Eldar, Nov 01 2024 *)

Crossrefs

Cf. A000203, A001008/A002805, A057641.

Sequence in context: A371185 A276112 A228075 * A309405 A354300 A032420

Adjacent sequences: A377567 A377568 A377569 * A377571 A377572 A377573

Keyword

nonn

Author

Ahmad J. Masad, Nov 01 2024

Status

approved