A067698 -id:A067698 - cp's OEIS frontend

A350299 Numbers k > 1 with sigma(k)/(k * log(log(k))) > sigma(m)/(m * log(log(m))) for all m > k, sigma(k) being A000203(k), the sum of the divisors of k.

3, 4, 6, 12, 24, 60, 120, 180, 360, 2520, 5040

Offset: 1

Thomas Strohmann, Dec 23 2021

nonn

Gronwall's theorem says that lim sup_{k -> infinity} sigma(k)/(k*log(log(k))) = exp(gamma). Moreover if the Riemann hypothesis is true, we have sigma(k)/(k*log(log(k))) < exp(gamma) when k > 5040 (gamma = Euler-Mascheroni constant).

The terms in the sequence listed above are provably correct since their ratios: sigma(k)/(k * log(log(k))) are greater than exp(gamma).

Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.

Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), pp. 251-256.
S. Nazardonyavi and S. Yakubovich, Extremely Abundant Numbers and the Riemann Hypothesis, Journal of Integer Sequences, 17 (2014), Article 14.2.8.
Thomas Strohmann, C++ code

Cf. A000203, A067698, A004394, A002093, A004490.

Cf. also A001620, A073004.

A353076 Odd positive integers k such that sigma(k) > exp(gamma) * k * log(log(k))/2.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 33, 35, 39, 45, 51, 55, 57, 63, 65, 69, 75, 81, 87, 93, 99, 105, 117, 135, 147, 153, 165, 171, 189, 195, 207, 225, 231, 255, 273, 285, 297, 315, 345, 351, 357, 375, 399, 405, 435, 441, 465, 495, 525, 555, 567, 585

Offset: 1

Amiram Eldar, Apr 22 2022

nonn

The first 23 oddly colossally abundant numbers (A110464) are in this sequence.

According to a proof by Washington and Yang (2021), the Riemann hypothesis is equivalent to the statement that all the terms of this sequence are smaller than A110464(24) = 18565284664427130919514350125.

			3 is in the sequence since 3 is odd and sigma(3) = 4 > exp(gamma) * 3 * log(log(3))/2 = 0.251... .

Amiram Eldar, Table of n, a(n) for n = 1..470
Lawrence C. Washington and Ambrose Yang, Analogues of the Robin-Lagarias criteria for the Riemann hypothesis, International Journal of Number Theory, Vol. 17, No. 4 (2021), pp. 843-870; arXiv preprint, arXiv:2008.04787 [math.NT], 2020.

Cf. A000203 (sigma), A067698, A073004 (exp(gamma)), A110464.

Select[Range[3, 600, 2], DivisorSigma[1, #] > Exp[EulerGamma] * # * Log[Log[#]]/2 &]

from sympy import divisor_sigma, EulerGamma, E, log
print([k for k in range(3, 600, 2) if divisor_sigma(k) > (E**EulerGamma * k * log(log(k)) / 2)]) # Karl-Heinz Hofmann, Apr 22 2022

A357330 Decimal expansion of sigma(N) / (N * log(log(N))) for N = 5040, where sigma = A000203.

Original entry on oeis.org

1, 7, 9, 0, 9, 7, 3, 3, 6, 6, 5, 3, 4, 8, 8, 1, 1, 3, 3, 3, 6, 1, 9, 0, 1, 3, 5, 0, 5, 9, 1, 0, 9, 5, 1, 7, 4, 0, 9, 0, 9, 5, 3, 9, 0, 7, 9, 8, 7, 5, 7, 3, 5, 7, 7, 9, 1, 7, 4, 6, 5, 3, 5, 2, 3, 5, 6, 6, 7, 0, 4, 6, 9, 5, 5, 7, 6, 9, 5, 2, 2, 9, 7, 7, 9, 3, 4, 2, 3, 5

Offset: 1

Jianing Song, Sep 24 2022

It is known that the Riemann Hypothesis (RH) is true if and only if sigma(n) < exp(gamma) * n * log(log(n)) for all n > 5040, where gamma = A001620 is the Euler-Mascheroni constant; that is to say, the RH is true if and only if 5040 is the last term in A067698.

			sigma(5040) / (5040 * log(log(5040))) = 1.79097336653488113336... In comparison, exp(gamma) = 1.78107241799019798523...

Eric Weisstein's World of Mathematics, Robin's Theorem.
Wikipedia, Riemann hypothesis.

Cf. A067698, A073004, A000203, A001620, A357331.

RealDigits[DivisorSigma[-1, 5040] / Log[Log[5040]], 10, 120][[1]] (* Amiram Eldar, Jun 19 2023 *)

PARI
```
sigma(5040) / (5040 * log(log(5040)))
```

Equals 403 / (105 * log(log(5040))).

A357331 Decimal expansion of sigma(N) / (exp(gamma) * N * log(log(N))) for N = 5040, where sigma = A000203 and gamma = A001620 is the Euler-Mascheroni constant.

Original entry on oeis.org

1, 0, 0, 5, 5, 5, 8, 9, 8, 1, 4, 5, 6, 7, 2, 0, 1, 0, 3, 6, 4, 2, 4, 7, 0, 7, 6, 7, 7, 8, 1, 5, 5, 4, 4, 3, 1, 6, 9, 8, 4, 4, 3, 0, 1, 4, 6, 7, 4, 1, 5, 2, 7, 9, 7, 3, 6, 8, 0, 2, 5, 8, 3, 2, 2, 5, 7, 4, 6, 5, 9, 5, 4, 9, 5, 5, 8, 5, 2, 2, 7, 8, 7, 7, 1, 4, 6, 2, 3, 9

Offset: 1

Jianing Song, Sep 24 2022

It is known that the Riemann Hypothesis (RH) is true if and only if sigma(n) < exp(gamma) * n * log(log(n)) for all n > 5040, where gamma = A001620 is the Euler-Mascheroni constant; that is to say, the RH is true if and only if 5040 is the last term in A067698.

			sigma(5040) / (exp(gamma) * 5040 * log(log(5040))) = 1.00555898145672010364... > 1.

Eric Weisstein's World of Mathematics, Robin's Theorem.
Wikipedia, Riemann hypothesis.

Cf. A067698, A073004, A000203, A001620, A357330.

RealDigits[DivisorSigma[-1, 5040] / (Exp[EulerGamma] * Log[Log[5040]]), 10, 120][[1]] (* Amiram Eldar, Jun 19 2023 *)

sigma(5040) / (exp(Euler) * 5040 * log(log(5040)))

Equals 403 / (exp(gamma) * 105 * log(log(5040))).

A211385 Values of n for which product_{p|n, p prime} 1 + 1/p > e^gamma*log(log(n)).

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 10, 12, 18, 30

Offset: 1

Arkadiusz Wesolowski, Feb 07 2013

nonn

30 is the last term:
- if and only if the Riemann hypothesis is true
- for which sigma(n) > tau(n)*phi(n)
- which appears in A060735

Eric W. Weisstein, MathWorld: Riemann Hypothesis

Cf. A001615, A060735, A064374, A067698.

lst = {}; Do[If[Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]) > E^EulerGamma*Log@Log[n], AppendTo[lst, n]], {n, 2, 30}]; lst

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 *)

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A350299 Numbers k > 1 with sigma(k)/(k * log(log(k))) > sigma(m)/(m * log(log(m))) for all m > k, sigma(k) being A000203(k), the sum of the divisors of k.

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A353076 Odd positive integers k such that sigma(k) > exp(gamma) * k * log(log(k))/2.

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A357330 Decimal expansion of sigma(N) / (N * log(log(N))) for N = 5040, where sigma = A000203.

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A357331 Decimal expansion of sigma(N) / (exp(gamma) * N * log(log(N))) for N = 5040, where sigma = A000203 and gamma = A001620 is the Euler-Mascheroni constant.

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A211385 Values of n for which product_{p|n, p prime} 1 + 1/p > e^gamma*log(log(n)).

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

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Author

Keywords

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Mathematica