A079526 -id:A079526 - cp's OEIS frontend

A259632 a(n) = floor(exp(H_k)*log(H_k)) - sigma(k) where k is the n-th colossally abundant number (Sequence A079526 applied to the colossally abundant numbers (A004490).)

-2, -2, -3, -2, 0, 28, 199, 483, 9040, 143814, 306295, 963844, 5155067, 81053615, 1334916470, 29106956400, 58655156000, 1817551640000, 56466287000000, 376943530000000, 1144451930000000, 41803527000000000

Offset: 1

Gene Ward Smith, Dec 17 2016

sign

It follows easily from the work of Lagarias that the Riemann hypothesis is equivalent to this sequence's being nonnegative except for the first four terms.

J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543.

Cf. A004490, A079526.

A057641 a(n) = floor(H(n) + exp(H(n))*log(H(n))) - sigma(n), where H(n) = Sum_{k=1..n} 1/k and sigma(n) (A000203) is the sum of the divisors of n.

Original entry on oeis.org

0, 0, 1, 0, 4, 0, 7, 2, 7, 5, 13, 0, 17, 9, 12, 8, 23, 5, 27, 8, 21, 20, 34, 1, 33, 25, 30, 17, 46, 7, 50, 22, 40, 37, 46, 6, 62, 43, 50, 19, 70, 19, 74, 37, 46, 55, 82, 9, 79, 46, 70, 47, 95, 32, 83, 38, 81, 74, 107, 2, 112, 81, 76, 56, 102, 45, 125, 70, 103, 58, 133, 14, 138, 101

Offset: 1

N. J. A. Sloane, Oct 12 2000

Theorem (Lagarias): a(n) is nonnegative for all n if and only if the Riemann Hypothesis is true.
Up to rank n=10^4, zeros occur only at n=1,2,4,6 and 12; ones occur at n=3 and n=24. The first occurrence of k = 0,1,2,3,... is at n = 1,3,8,-1,5,10,36,7,16,14,-1,-1,15,11,72,... where -1 means that k does not occur among the first 10^4 terms. - Robert G. Wilson v, Dec 06 2010, reformulated by M. F. Hasler, Sep 09 2011
Looking at the graph of this sequence, it appears that there is a slowly growing lower bound. It is even more apparent when larger ranges of points are computed. Numbers A176679(n+2) and A222761(n) give the (x,y) coordinates of the n-th point. - T. D. Noe, Mar 28 2013

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

Peter Luschny, Table of n, a(n) for n = 1..20000 (first 10000 terms from T. D. Noe)
Masazumi Honda and Takuya Yoda, String theory, N = 4 SYM and Riemann hypothesis, arXiv:2203.17091 [hep-th], 2022.
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.
S. Nazardonyavi and S. Yakubovich, Delicacy of the Riemann hypothesis and certain subsequences of superabundant numbers, arXiv preprint arXiv:1306.3434 [math.NT], 2013.

Cf. A057640, A000203, A076633, A067698, A079526, A058209.

f[n_] := Block[{h = HarmonicNumber@n}, Floor[h + Exp@h*Log@h] - DivisorSigma[1, n]]; Array[f, 74] (* Robert G. Wilson v, Dec 06 2010 *)

a(n)={my(H=sum(k=1,n,1/k)); floor(exp(H)*log(H)+H) - sigma(n)}
list_A057641(Nmax,H=0,S=1)=for(n=S,Nmax, H+=1/n; print1(floor(exp(H)*log(H)+H) - sigma(n),","))  \\ M. F. Hasler, Sep 09 2011

a(n) = A057640(n) - A000203(n). - Omar E. Pol, Oct 25 2019

Five more terms from Robert G. Wilson v, Dec 06 2010

I deleted some unproved assertions by Robert G. Wilson v about the presence of 0's, 1's, ... in this sequence. - N. J. A. Sloane, Dec 07 2010

A079527 a(n) = floor( exp(H_n)*log(H_n) ).

Original entry on oeis.org

0, 1, 3, 5, 8, 10, 12, 15, 17, 20, 22, 25, 27, 30, 33, 35, 38, 41, 43, 46, 49, 52, 55, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 86, 89, 92, 95, 98, 101, 104, 107, 110, 113, 116, 119, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 160, 163, 166, 169, 172, 175, 178

Offset: 1

N. J. A. Sloane, Jan 22 2003

nonn

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).

Cf. A079526.

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

a[n_] := Exp[HarmonicNumber[n]] Log[HarmonicNumber[n]] // Floor;
Array[a, 64] (* Jean-François Alcover, Oct 08 2018 *)

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

[floor(exp(harmonic_number(n))*log(harmonic_number(n))) 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

A259632 a(n) = floor(exp(H_k)*log(H_k)) - sigma(k) where k is the n-th colossally abundant number (Sequence A079526 applied to the colossally abundant numbers (A004490).)

Views

Author

Keywords

Comments

Links

Crossrefs

A057641 a(n) = floor(H(n) + exp(H(n))*log(H(n))) - sigma(n), where H(n) = Sum_{k=1..n} 1/k and sigma(n) (A000203) is the sum of the divisors of n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A079527 a(n) = floor( exp(H_n)*log(H_n) ).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica