A057640 -id:A057640 - cp's OEIS frontend

A328363 Partial sums of A057640.

1, 4, 9, 16, 26, 38, 53, 70, 90, 113, 138, 166, 197, 230, 266, 305, 346, 390, 437, 487, 540, 596, 654, 715, 779, 846, 916, 989, 1065, 1144, 1226, 1311, 1399, 1490, 1584, 1681, 1781, 1884, 1990, 2099, 2211, 2326, 2444, 2565, 2689, 2816, 2946, 3079, 3215, 3354, 3496, 3641, 3790, 3942, 4097, 4255, 4416

Offset: 1

Omar E. Pol, Oct 26 2019

nonn

Cf. A000203, A001008, A002805, A024916, A057640, A057641, A328367.

f:=func; [&+[f(k):k in [1..n]]:n in [1..57]]; // Marius A. Burtea, Nov 18 2019

a(n) = A024916(n) + A328367(n).

A002410 Nearest integer to imaginary part of n-th zero of Riemann zeta function.

Original entry on oeis.org

14, 21, 25, 30, 33, 38, 41, 43, 48, 50, 53, 56, 59, 61, 65, 67, 70, 72, 76, 77, 79, 83, 85, 87, 89, 92, 95, 96, 99, 101, 104, 105, 107, 111, 112, 114, 116, 119, 121, 123, 124, 128, 130, 131, 133, 135, 138, 140, 141, 143, 146, 147, 150, 151, 153, 156, 158, 159, 161

Offset: 1

N. J. A. Sloane

"All these zeros of the form s + it have real part s = 1/2 and are simple. Thus the Riemann hypothesis is true at least for t < 3330657430697." - Wedeniwski
From Daniel Forgues, Jul 24 2009: (Start)
All nontrivial zeros on the critical line, of the form 1/2 + i*t, have an associated conjugate nontrivial zero of the form 1/2 - i*t.
Any nontrivial zeros off the critical line, if ever found, would come in pairs (1/2 +- delta) + i*t, 0 < delta < 1/2. Each of these pairs, again if ever found, would then have their associated conjugate pair (1/2 +- delta) - i*t, 0 < delta < 1/2. (End)
The sequence is not strictly increasing. - Joerg Arndt, Jan 17 2015
The fraction of numbers n such that a(n) = a(n-1) has density 1. There are only finitely many numbers n with a(n) > a(n-1) + 1, see A208436. - Charles R Greathouse IV, Mar 07 2018
Conjecture: Noninteger rationals of the form m/2^bigomega(m) that can be used to approximate this sequence, i.e. a(n) ~~ 2*Pi*A374074(n)/2^bigomega(A374074(n)) - n/2 +- (...), where '~~' means 'close to'. - Friedjof Tellkamp, Jul 04 2024

			The imaginary parts of the first 4 zeros are 14.134725... (A058303), 21.0220396... (A065434), 25.01085758... (A065452), 30.424876... (A065453).

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W. F. Galway, Computations related to the Riemann Hypothesis
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D. A. Goldston & S. M. Gonek, A note on S(T) and the zeros of the Riemann zeta-function, arXiv:math/0511092 [math.NT], 2005.
S. Gonek, Three Lectures on the Riemann Zeta-Function, arXiv:math/0401126 [math.NT], 2004.
J. Good & B. Churchhouse, A New Conjecture Related to the Riemann Hypothesis
X. Gourdon, The 10^13 first zeros of the Riemann Zeta function and zeros computation at very large height
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S. W. Graham, Review of "Prime Obsession" by J. Derbyshire
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J. P. Gram, Note sur les zéros de la fonction zeta(s) de Riemann, Acta Mathematica, 27 (1903), 289-304.
Mats Granvik, How plot the Riemann zeta zero spectrum with the Fourier transform in Mathematica?
Mats Granvik, Illustration of Andre LeClair's approximation of the initial terms
Mats Granvik, Mathematica programs comparing limiting ratios vs Newton-Raphson method for finding Riemann zeta zeros.
Mats Granvik, Plots from the Mathematica programs comparing limiting ratios vs Newton-Raphson method for finding Riemann zeta zeros.
A. Granville, Nombres premiers et chaos quantique (Text in French)
J. Hadamard, Sur la distribution des zeros de la fonction zeta(s) et ses consequences arithmetiques (Text in French)
B. Hayes, The Spectrum of Riemannium
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A. Ivic & H. J. J. te Riele, On the zeros of the error term for the mean square of | zeta(1/2 + it) |
D. Jao, PlanetMath.Org, Riemann zeta function
C.-X. Jiang, Disproofs Of Riemann's Hypothesis
N. M. Katz & P. Sarnak, Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. 36 (1999), 1-26.
E. Klarreich, Prime Time
A. F. Lavrik, Riemann hypotheses
A. LeClair, An electrostatic depiction of the validity of the Riemann Hypothesis and a formula for the N-th zero at large N, arXiv:1305.2613 [math-ph], 2013.
N. Levinson, At Least One-Third of Zeros of Riemann's Zeta-Function are on sigma=1/2, Proc Natl Acad Sci U S A. 1974 Apr; 71(4): 1013-1015.
P. Li, On Montgomery's Pair Correlation Conjecture to the Zeros of the Riemann Zeta Function
Lionman & Allispaul, Riemann Hypothesis
LMDBF, The Riemann zeta-function
J. van de Lune, H. J. J. te Riele & D. T. Winter, Rigorous High Speed Separation Of Zeros Of Riemann's Zeta Function
J. van de Lune & H. J. J. te Riele, Rigorous High Speed Separation Of Zeros Of Riemann's Zeta Function, II
J. van de Lune & H. J. J. te Riele, Rigorous High Speed Separation Of Zeros of Riemann's Zeta Function, III
J. van de Lune & H. J. J. te Riele, On the Zeros of the Riemann Zeta Function in the Critical Strip, III
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A. M. Odlyzko, Primes, Quantum Chaos and Computers
A. M. Odlyzko, On The Distribution Of Spacings Between Zeros Of The Zeta Function
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A. M. Odlyzko & M. Schoenhage, Fast algorithms for multiple evaluations of the Riemann zeta function
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Ed. Pegg Jr., The Riemann Hypothesis
J. C. Perez-Moure, Evidences that the Riemann Hypothesis is true
Simon Plouffe, The first (non trivial) zeros of the Riemann Zeta function
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H. J. J. te Riele, Some historical and other notes about the Mertens conjecture and its recent disproof
H. J. J. te Riele, On the History of the function M(x)/sqrt(x) since Stieltjes
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Jörn Steuding, On simple zeros of the Riemann zeta-function
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Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros
Eric Weisstein's World of Mathematics, Xi-Function
Wikipedia, Riemann hypothesis
Wolfram Research, First few computations of Z(n) (nontrivial zeros power sums)
A. Zaccagnini, Primes in almost all short intervals and the distribution of the zeros of the Riemann zeta-function
Index entries for zeta function

Cf. A013629 (floor), A092783 (ceiling), A057641, A057640, A058209, A058210, A120401, A122526, A072080, A124288 ("unstable" zeta zeros), A124289 ("unstable twins"), A236212, A177885, A374074 (approximation).

Imaginary part of k-th nontrivial zero of Riemann zeta function: A058303 (k=1), A065434 (k=2), A065452 (k=3), A065453 (k=4), A192492 (k=5), A305741 (k=6), A305742 (k=7), A305743 (k=8), A305744 (k=9), A306004 (k=10).

Table[Round[Im[ZetaZero[n]]], {n, 59}] (* Jean-François Alcover, May 02 2011 *)

apply(round,lfunzeros(lzeta,100)) \\ Charles R Greathouse IV, Mar 10 2016

def A002410_list(n):
    Z = lcalc.zeros(n)
    return [round(z) for z in Z]
A002410_list(59) # Peter Luschny, May 02 2014

a(n) ~ (2*Pi*e) * e^(W0(n/e)), where W0 is the principal branch of Lambert's W function. - Charles R Greathouse IV, Sep 14 2012, corrected by Hal M. Switkay, Oct 04 2021

a(n) ~ 2*Pi*(n - 11/8)/ProductLog((n - 11/8)/exp(1)). This is the asymptotic by Guilherme França and André LeClair. - Mats Granvik, Mar 10 2015; corrected May 16 2016

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 08 2004

A058303 Decimal expansion of the imaginary part of the first nontrivial zero of the Riemann zeta function.

Original entry on oeis.org

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

Offset: 2

Robert G. Wilson v, Dec 08 2000

"The Riemann Hypothesis, considered by many to be the most important unsolved problem of mathematics, is the assertion that all of zeta's nontrivial zeros line up with the first two all of which lie on the line 1/2 + sqrt(-1)*t, which is called the critical line. It is known that the hypothesis is obeyed for the first billion and a half zeros." (Wagon)

We can compute 105 digits of this zeta zero as the numerical integral: gamma = Integral_{t=0..gamma+15} (1/2)*(1 - sign((RiemannSiegelTheta(t) + Im(log(zeta(1/2 + i*t))))/Pi - n + 3/2)) where n=1 and where the initial value of gamma = 1. The upper integration limit is arbitrary as long as it is greater than the zeta zero computed recursively. The recursive formula fails at zeta zeros with indices n equal to sequence A153815. - Mats Granvik, Feb 15 2017

			14.1347251417346937904572519835624702707842571156992...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.15.3, p. 138.
S. Wagon, "Mathematica In Action," W. H. Freeman and Company, NY, 1991, page 361.

Iain Fox, Table of n, a(n) for n = 2..20000
P. J. Forrester and A. Mays, Finite size corrections in random matrix theory and Odlyzko's data set for the Riemann zeros, arXiv preprint arXiv:1506.06531 [math-ph], 2015.
P. J. Forrester and A. Mays, Finite size corrections in random matrix theory and Odlyzko's data set for the Riemann zeros, Proceedings of the Royal Society A, Vol: 471, Issue: 2182, 2015.
Fredrik Johansson, The first nontrivial zero to over 300000 decimal digits.
Andrew M. Odlyzko, The first 100 (non trivial) zeros of the Riemann Zeta function, to over 1000 decimal digits each, AT&T Labs - Research.
Andrew M. Odlyzko, Tables of zeros of the Riemann zeta function.
Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros.
Eric Weisstein's World of Mathematics, Xi-Function.
Index entries for zeta function.

Imaginary part of k-th nontrivial zero of Riemann zeta function: A058303 (k=1: this), A065434 (k=2), A065452 (k=3), A065453 (k=4), A192492 (k=5), A305741 (k=6), A305742 (k=7), A305743 (k=8), A305744 (k=9), A306004 (k=10).

Cf. A002410 (round), A013629 (floor); A057641, A057640, A058209, A058210.

Digits:= 150; Re(fsolve(Zeta(1/2+I*t)=0, t=14.13)); # Iaroslav V. Blagouchine, Jun 24 2016

FindRoot[ Zeta[1/2 + I*t], {t, 14 + {-.3, +.3}}, AccuracyGoal -> 100, WorkingPrecision -> 120]
RealDigits[N[Im[ZetaZero[1]], 100]][[1]] (* Charles R Greathouse IV, Apr 09 2012 *)
(* The following numerical integral takes about 9 minutes to compute *)Clear[n, t, gamma]; gamma = 1; numberofzetazeros = 1; Quiet[Do[gamma = N[NIntegrate[(1/2)*(1 - Sign[(RiemannSiegelTheta[t] + Im[Log[Zeta[I*t + 1/2]]])/Pi - n + 3/2]), {t, 0, gamma + 15}, PrecisionGoal -> 110, MaxRecursion -> 350, WorkingPrecision -> 120], 105]; Print[gamma], {n, 1, numberofzetazeros}]]; RealDigits[gamma][[1]] (* Mats Granvik, Feb 15 2017 *)

solve(x=14,15,imag(zeta(1/2+x*I))) \\ Charles R Greathouse IV, Feb 26 2012

lfunzeros(1,15)[1] \\ Charles R Greathouse IV, Mar 07 2018

zeta(1/2 + i*14.1347251417346937904572519836...) = 0.

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

A076633 Incorrect guess for index of n-th local maxima (in decreasing order) of f(k) = (sigma(k) - H_k)/(exp(H_k)log(H_k)), where H_k = 1 + 1/2 + 1/3 + ... + 1/k.

Original entry on oeis.org

12, 120, 60, 2520, 5040, 360, 24, 840, 55440, 10080

Offset: 1

Luke Pebody (pebodyl(AT)msci.memphis.edu), Oct 22 2002

nonn

Lagarias showed that the Riemann Hypothesis is equivalent to the formula sigma(k) <= H_k + exp(H_k)log(H_k) for all k >= 1 with equality only when k=1. In other words f(k)<1 for all k. At first glance it seems that f(12) is the largest value of f, followed by f(120), f(60) and so on. Proving that f(12) is indeed the largest value would prove the Riemann Hypothesis. However, f(12) is not the largest value.

The terms shown are merely the maxima for "small" values of k. If the function f(k) is evaluated at colossally abundant numbers (A004490), we find that beyond the 58th colossally abundant number, which is over 10^76, the function is greater than f(12) and increasing at each subsequence colossally abundant number. Use A073751 to generate colossally abundant numbers not in A004490. - T. D. Noe, Oct 24 2002

J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543.
T. D. Noe, Plot of the function f for the first 200 colossally abundant numbers

Cf. A057640, A057641, A004490, A073751.

A353149 Sum of the odd-indexed terms in the n-th row of the triangle A196020.

Original entry on oeis.org

1, 3, 5, 7, 9, 12, 13, 15, 20, 19, 21, 28, 25, 27, 37, 31, 33, 44, 37, 42, 52, 43, 45, 60, 54, 51, 68, 56, 57, 83, 61, 63, 84, 67, 81, 92, 73, 75, 100, 90, 81, 113, 85, 87, 130, 91, 93, 124, 104, 114, 132, 103, 105, 143, 126, 120, 148, 115, 117, 175, 121, 123, 180, 127, 150, 173, 133, 135, 180, 175

Offset: 1

Omar E. Pol, Apr 26 2022

nonn

a(n) is the total number of steps in all odd-indexed double-staircases of the diagram of A196020 with n levels (see the example).
a(n) is also the total number of steps in all odd-indexed double-staircases of the diagram described in A335616 with n levels that have at least one step in the bottom level of the diagram.
Sigma(n) <= a(n).
The graph of the sum-of-divisors function A000203 is intermediate between the graph of this sequence and the graph of A353154 (see link). - Omar E. Pol, May 13 2022

			For n = 15 the 15th row of the triangle A196020 is [29, 13, 7, 0, 1]. The sum of the odd-indexed terms is 29 + 7 + 1 = 37, so a(15) = 37.
Illustration of a(15) = 37:
Level                                   Diagram
.                                          _
1                                        _|1|_
2                                      _|1   1|_
3                                    _|1       1|_
4                                  _|1           1|_
5                                _|1       _       1|_
6                              _|1        |1|        1|_
7                            _|1          | |          1|_
8                          _|1           _| |_           1|_
9                        _|1            |1   1|            1|_
10                     _|1              |     |              1|_
11                   _|1               _|     |_               1|_
12                 _|1                |1       1|                1|_
13               _|1                  |         |                  1|_
14             _|1                   _|    _    |_                   1|_
15            |1                    |1    |1|    1|                    1|
.
The diagram has 37 steps, so a(15) = 37.

David A. Corneth, Table of n, a(n) for n = 1..10000
OEIS Plot 2, A353149 vs A000203

Cf. A000203, A057640, A196020, A209246, A211343, A235791, A236104, A237591, A237593, A285901, A335616, A347186, A353154.

a(n) = { my(r = A196020row(n)); sum(i = 0, (#r-1)\2, r[2*i + 1]) }
A196020row(n) = { my(res, qc); qc = (sqrtint(8*n + 1) - 1)\2; res = vector(qc); for(i = 1, qc, cn = n - binomial(i + 1, 2); if(cn % i == 0, res[i] = 2*(cn/i) + 1 ) ); res } \\ David A. Corneth, Apr 28 2022

a(n) = A000203(n) + A353154(n).

a(n) = A209246(n) - A353154(n).

A328367 Partial sums of A057641.

Original entry on oeis.org

0, 0, 1, 1, 5, 5, 12, 14, 21, 26, 39, 39, 56, 65, 77, 85, 108, 113, 140, 148, 169, 189, 223, 224, 257, 282, 312, 329, 375, 382, 432, 454, 494, 531, 577, 583, 645, 688, 738, 757, 827, 846, 920, 957, 1003, 1058, 1140, 1149, 1228, 1274, 1344, 1391, 1486, 1518, 1601, 1639, 1720, 1794, 1901, 1903, 2015, 2096

Offset: 1

Omar E. Pol, Oct 26 2019

nonn

Cf. A000203, A001008, A002805, A024916, A057640, A057641, A328363.

f:=func; [&+[f(k):k in [1..n]]:n in [1..62]]; // Marius A. Burtea, Nov 18 2019

a(n) = A328363(n) - A024916(n).

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.

Original entry on oeis.org

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

Offset: 1

Peter Luschny, Oct 15 2020

Conjecture: This sequence is finite.

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.

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.

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

Cf. A001008, A002805.

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 *)

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

A353130 a(n) = floor(H(n) + exp(H(n))*log(H(n))) - n, where H(n) = Sum_{k=1..n} 1/k.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 9, 11, 13, 14, 16, 18, 19, 21, 23, 24, 26, 28, 30, 32, 34, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 96, 98, 100, 102, 104, 106, 108, 110, 113, 115, 117, 119, 121, 123, 126, 128

Offset: 1

Omar E. Pol, Apr 24 2022

nonn

About Lagarias's theorem and the Riemann hypothesis the graph of A057640 vs. A000203 is essentially equivalent to the graph of this sequence vs. A001065 (see Plot 2 in the Links section and A057640, A057641).

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.
OEIS Plot 2, A353130 vs A001065

Cf. A000203, A057640, A057641, A001065.

a[n_] := Module[{h = HarmonicNumber[n]}, Floor[h + Exp[h]*Log[h]] - n]; Array[a, 100] (* Amiram Eldar, Apr 26 2022 *)

H(n) = sum(k=1, n, 1/k)
a(n) = floor(H(n) + exp(H(n))*log(H(n))) - n \\ Felix Fröhlich, Apr 26 2022

a(n) = A057640(n) - n.

a(n) = A057641(n) + A001065(n).

cp's OEIS Frontend

A328363 Partial sums of A057640.

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A002410 Nearest integer to imaginary part of n-th zero of Riemann zeta function.

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A058303 Decimal expansion of the imaginary part of the first nontrivial zero of the Riemann zeta function.

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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.

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A076633 Incorrect guess for index of n-th local maxima (in decreasing order) of f(k) = (sigma(k) - H_k)/(exp(H_k)log(H_k)), where H_k = 1 + 1/2 + 1/3 + ... + 1/k.

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A353149 Sum of the odd-indexed terms in the n-th row of the triangle A196020.

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A328367 Partial sums of A057641.

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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.

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