A079527 -id:A079527 - cp's OEIS frontend

A079526 a(n) = floor( exp(H_n)*log(H_n) ) - sigma(n).

-1, -2, -1, -2, 2, -2, 4, 0, 4, 2, 10, -3, 13, 6, 9, 4, 20, 2, 23, 4, 17, 16, 31, -3, 29, 21, 26, 13, 42, 3, 46, 18, 36, 32, 41, 1, 57, 38, 45, 14, 65, 14, 69, 32, 41, 51, 78, 5, 75, 42, 66, 43, 90, 27, 78, 33, 76, 70, 103, -2, 107, 76, 71, 51, 98, 41, 120, 65, 98, 53

Offset: 1

N. J. A. Sloane, Jan 22 2003

sign

M. Kaneko has shown that the Riemann hypothesis is equivalent to the assertion that a(n) > 0 for n > 60.

G. C. Greubel, Table of n, a(n) for n = 1..10000
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.

H_n = sum of harmonic series (see A002387), sigma(n) = A000203.

Cf. A057641, A058209, A079527.

[Floor(Exp(HarmonicNumber(n))*Log(HarmonicNumber(n))) - DivisorSigma(1,n): n in [1..80]]; // G. C. Greubel, Jan 15 2019

f[n_] := Floor[Exp[HarmonicNumber[n]]Log[HarmonicNumber[n]]] - DivisorSigma[1, n]; Array[f, 70] (* Robert G. Wilson v, Dec 17 2016 *)

{h(n) = sum(k=1, n, 1/k)};
vector(80, n, floor( exp(h(n))*log(h(n))) - sigma(n,1) ) \\ G. C. Greubel, Jan 15 2019

[floor(exp(harmonic_number(n))*log(harmonic_number(n))) - sigma(n,1) for n in (1..80)] # G. C. Greubel, Jan 15 2019

A306348 Numbers k such that exp(H_k)*log(H_k) <= sigma(k), where H_k is the harmonic number.

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 24, 60

Offset: 1

Seiichi Manyama, Feb 09 2019

If the Riemann hypothesis is true, there are no more terms.

			Let b(n) = exp(H_{a(n)})*log(H_{a(n)}).
n | a(n) |    b(n)    | sigma(a(n))
--+------+------------+-------------
1 |   1  |   0        |      1
2 |   2  |   1.817... |      3
3 |   3  |   3.791... |      4
4 |   4  |   5.894... |      7
5 |   6  |  10.384... |     12
6 |  12  |  25.218... |     28
7 |  24  |  57.981... |     60
8 |  60  | 166.296... |    168

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.

Cf. A000203, A067698, A079526, A079527.

For[k = 1, True, k++, If[Exp[HarmonicNumber[k]] Log[HarmonicNumber[k]] <= DivisorSigma[1, k], Print[k]]] (* Jean-François Alcover, Feb 14 2019 *)

cp's OEIS Frontend

A079526 a(n) = floor( exp(H_n)*log(H_n) ) - sigma(n).

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