A073004 -id:A073004 - cp's OEIS frontend

A002389 Decimal expansion of -log(gamma), where gamma is Euler's constant A001620.

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

Offset: 0

N. J. A. Sloane

From Peter Bala, Aug 24 2025: (Start)
By definition, the Euler-Mascheroni constant gamma = lim_{n -> oo} s(n), where s(n) = Sum_{k = 1..n} 1/k - log(n). The convergence is slow. For example, s(50) = 0.5(87...) is only correct to 1 decimal digit. Let S(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*s(n+k). Elsner shows that S(n) converges to gamma much more rapidly. For example, S(50) = 0.57721566490153286060651209008(02...) gives gamma correct to 29 decimal digits.
Define E(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*log(s(n+k)). Then it appears that E(n) converges rapidly to log(gamma). For example, E(50) = -0.549539312981644822337661768802(88...) gives log(gamma) correct to 30 decimal digits. Cf. A073004. (End)

			.549539312981644822337661768802907788330698981263...

W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. XVIII.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ivan Panchenko, Table of n, a(n) for n = 0..1000
C. Elsner, On a sequence transformation with integral coefficients for Euler's constant, Proc. Amer. Math. Soc., Vol. 123 (1995), Number 5, pp. 1537-1541.
Simon Plouffe, -log(gamma) to 10000 digits
Simon Plouffe, -log(gamma) to 1024 digits

Cf. A001620, A073004, A155969, A213440.

SetDefaultRealField(RealField(100)); R:= RealField(); -Log(EulerGamma(R)); // G. C. Greubel, Sep 07 2018

RealDigits[-Log[EulerGamma], 10, 100][[1]] (* G. C. Greubel, Sep 07 2018 *)

-log(Euler) \\ Michel Marcus, Mar 11 2013

A094644 Continued fraction for e^gamma.

Original entry on oeis.org

1, 1, 3, 1, 1, 3, 5, 4, 1, 1, 2, 2, 1, 7, 9, 1, 16, 1, 1, 1, 2, 6, 1, 2, 1, 6, 2, 59, 1, 1, 1, 3, 3, 3, 2, 1, 3, 5, 100, 1, 58, 1, 2, 1, 94, 1, 1, 2, 2, 10, 1, 2, 7, 1, 3, 4, 5, 3, 10, 1, 21, 1, 11, 1, 4, 1, 2, 2, 1, 2, 2, 1, 8, 3, 2, 1, 1, 6, 1, 2, 2, 1, 38, 2, 1, 4, 1, 3, 1, 1, 5, 3, 1, 52, 1, 2, 2, 1, 1

Offset: 0

Jonathan Sondow and Robert G. Wilson v, May 18 2004

Increasing partial quotients are: 1,3,5,7,9,16,59,100,129,314,2294,1568705

e^gamma appears in theorems of Mertens, Gronwall, Ramanujan, and Robin on primes, the sum-of-divisors function, and the Riemann Hypothesis (see Caveney-Nicolas-Sondow 2011, pp. 1-2).

			1 + 1/(1 + 1/(3 + 1/(1 + 1/(1 + 1/(3 + 1/(5 + 1/(4 + ...)))))))

J. Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 97.
G. Boros and V. Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004, Chap. 10.

T. D. Noe, Table of n, a(n) for n = 0..9999 (444 terms from Bo Gyu Jeong)
G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), Article A33.
Jonathan Sondow, An antisymmetric formula for Euler's constant, Math. Mag. 71 (1998), 219-220.
Jonathan Sondow, An antisymmetric formula for Euler's constant, Math. Mag. 71 (1998), 219-220.
Jonathan Sondow, Criteria for irrationality of Euler's constant, Proc. Amer. Math. Soc. 131 (2003), 3335-3344.
Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, arXiv:math/0211148 [math.CA], 2002-2004.
Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, Amer. Math. Monthly 112 (2005), 61-65.
Jonathan Sondow, An infinite product for e^gamma via hypergeometric formulas for Euler's constant, gamma, arXiv:math/0306008 [math.CA], 2003.
Jonathan Sondow, A faster product for pi and a new integral for ln pi/2, arXiv:math/0401406 [math.NT], 2004.
Jonathan Sondow, A faster product for pi and a new integral for ln pi/2, Amer. Math. Monthly 112 (2005), 729-734 and 113 (2006), 670.
Jonathan Sondow and Sergey Zlobin, A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant, arXiv:math/0211075 [math.NT], 2002-2009.
Jonathan Sondow and Sergey Zlobin, A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant, Math. Slovaca 59 (2009), 1-8.
Jonathan Sondow and Wadim Zudilin, Euler's constant, q-logarithms and formulas of Ramanujan and Gosper, arXiv:math/0304021 [math.NT], 2003.
Jonathan Sondow and Wadim Zudilin, Euler's constant, q-logarithms and formulas of Ramanujan and Gosper, Ramanujan J. 12 (2006), 225-244.

Cf. A073004 = decimal expansion of exp(gamma).
Gamma is the Euler-Mascheroni constant A001620.
Cf. A079650 = continued fraction for exp(-gamma). [From R. J. Mathar, Sep 05 2008]

ContinuedFraction[ Exp[ EulerGamma], 100]

contfrac(exp(Euler)) \\ Amiram Eldar, Jun 13 2021

Offset changed by Andrew Howroyd, Aug 07 2024

A236435 Numerator of Product_{k=1..n-1} (1 + 1/prime(k)).

Original entry on oeis.org

1, 3, 2, 12, 96, 1152, 2304, 41472, 165888, 3981312, 119439360, 3822059520, 7644119040, 321052999680, 1284211998720, 61642175938560, 3328677500682240, 199720650040934400, 399441300081868800, 1597765200327475200, 115039094423578214400, 230078188847156428800, 18406255107772514304000

Offset: 1

Jonathan Sondow, Feb 01 2014

A236436(n)/(a(n)*zeta(2)) is the asymptotic density of the prime(n-1)-rough squarefree numbers (squarefree numbers whose prime factors are all >= prime(n-1)) for n >= 2. E.g., A236436(2)/(a(2)*zeta(2)) = 2/(3*zeta(2)) = 4/Pi^2 (A185199) is the asymptotic density of the odd squarefree numbers (A056911), and A236436(3)/(a(3)*zeta(2)) = 1/(2*zeta(2)) = 3/Pi^2 (A104141) is the asymptotic density of the 5-rough squarefree numbers (A276378). - Amiram Eldar, Aug 26 2025

			(1 + 1/2)*(1 + 1/3)*(1 + 1/5)*(1 + 1/7) = 96/35 has numerator a(5) = 96.
Fractions begin with 1, 3/2, 2, 12/5, 96/35, 1152/385, 2304/715, 41472/12155, 165888/46189, 3981312/1062347, 119439360/30808063, 3822059520/955049953, ...

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979; Theorem 429.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Jonathan Sondow and Eric W. Weisstein, MathWorld: Mertens Theorem.

Cf. A038110, A060753, A072044, A072045, A073004, A236436 (denominators), A335004.

Cf. A013661, A056911, A104141, A185199, A276378.

Numerator@Table[ Product[ 1 + 1/Prime[ k], {k, 1, n-1}], {n, 1, 23}]

a(n+1) / A236436(n+1) = (A072045(n)/A072044(n)) / (A038110(n+1)/A060753(n+1)) because 1+x = (1-x^2) / (1-x).

a(n) / A236436(n) = Product_{k=1..n-1} (1 + 1/prime(k)) ~ (6/Pi^2)*exp(gamma)*log(n) as n -> infinity, by Mertens's theorem.

A052488 a(n) = floor(n*H(n)) where H(n) is the n-th harmonic number, Sum_{k=1..n} 1/k (A001008/A002805).

Original entry on oeis.org

1, 3, 5, 8, 11, 14, 18, 21, 25, 29, 33, 37, 41, 45, 49, 54, 58, 62, 67, 71, 76, 81, 85, 90, 95, 100, 105, 109, 114, 119, 124, 129, 134, 140, 145, 150, 155, 160, 165, 171, 176, 181, 187, 192, 197, 203, 208, 214, 219, 224, 230, 235, 241, 247, 252, 258, 263, 269

Offset: 1

Tomas Mario Kalmar (TomKalmar(AT)aol.com), Mar 15 2000

Floor(n*H(n)) gives a (very) rough approximation to the n-th prime.

a(n) is the integer part of the solution to the Coupon Collector's Problem. For example, if there are n=4 different prizes to collect from cereal boxes and they are equally likely to be found, then the integer part of the average number of boxes to buy before the collection is complete is a(4)=8. - Ron Lalonde (ronronronlalonde(AT)hotmail.com), Feb 04 2004

John D. Barrow, One Hundred Essential Things You Didn't Know You Didn't Know, Ch. 3, 'On the Cards', W. W. Norton & Co., NY & London, 2008, pp. 30-32.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001008, A002805, A006218, A060293.

Cf. A001620, A073004.

[Floor(n*HarmonicNumber(n)): n in [1..60]]; // G. C. Greubel, May 14 2019

for n from 1 to 100 do printf(`%d,`,floor(n*sum(1/k, k=1..n))) od:
# Alternatively:
A052488:= n -> floor(n*(Psi(n+1)+gamma));
seq(A052488(n),n=1..100); # Robert Israel, May 19 2014

f[n_] := Floor[n*HarmonicNumber[n]]; Array[f, 60] (* Robert G. Wilson v, Nov 23 2015 *)

a(n) = floor(n*sum(k=1, n, 1/k)) \\ Altug Alkan, Nov 23 2015

from math import floor
n=100 #number of terms
ans=0
finalans = []
for i in range(1, n+1):
    ans+=(1/i)
    finalans.append(floor(ans*i))
print(finalans)
# Adam Hugill, Feb 14 2022

from fractions import Fraction
from itertools import count, islice
def agen():
    Hn = 0
    for n in count(1):
        Hn += Fraction(1, n)
        yield (n*Hn.numerator)//Hn.denominator
print(list(islice(agen(), 60))) # Michael S. Branicky, Aug 10 2022

from sympy import harmonic
def A052488(n): return int(n*harmonic(n)) # Chai Wah Wu, Oct 24 2023

[floor(n*harmonic_number(n)) for n in (1..60)] # G. C. Greubel, May 14 2019

More terms from James Sellers, Mar 17 2000

A091724 Decimal expansion of e^(2*EulerGamma).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Feb 01 2004

			3.17221895812545052772791340906947497712295773777230...

Robert Price, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Florian Luca, On the composition of the Euler function and the sum of divisors function, Colloquium Mathematicum, Vol. 108, No. 1 (2007), pp. 31-51.
Eric Weisstein's World of Mathematics, Exponential Integral.

Cf. A001620, A073004, A370689, A370690.

RealDigits[Exp[2*EulerGamma], 10, 100][[1]] (* Amiram Eldar, Jun 25 2021 *)

exp(2*Euler) \\ Michel Marcus, Jun 25 2021

Equals lim_{x -> 0} e^(2*ExpIntegralEi(-x))/x^2.
Equals A073004^2. - Michel Marcus, Jun 25 2021
Equals lim sup_{n->oo} H(n)/log_2(n)^2, where H(n) = A370689(n)/A370690(n) (De Koninck and Luca, 2007). - Amiram Eldar, Feb 27 2024

A227242 Decimal expansion of (e^gamma - 1)/e^gamma.

Original entry on oeis.org

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

Offset: 0

Arkadiusz Wesolowski, Oct 19 2013

The value is equal to lim_{n->oo} (Sum_{d|n#, d>n} 1/phi(d))/(Sum_{d|n#} 1/phi(d)).

			(exp(gamma) - 1)/exp(gamma) = 0.438540516433114830175856785....

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A000142, A001113, A001620, A073004, A080130, A351901.

E:=EulerGamma(RealField(105)); Reverse(Intseq(Floor(10^105*(Exp(E)-1)/Exp(E))));

evalf(1-exp(-gamma), 120);  # Alois P. Heinz, Feb 24 2022

RealDigits[(E^EulerGamma - 1)/E^EulerGamma, 10, 50][[1]] (* G. C. Greubel, Oct 02 2017 *)

default(realprecision, 105); x=10*(exp(Euler)-1)/exp(Euler); for(n=1, 105, d=floor(x); x=(x-d)*10; print1(d, ", "));

From Alois P. Heinz, Feb 24 2022: (Start)
Equals 1 - exp(-gamma) = 1 - A080130.
Equals lim_{n->oo} A351901(n)/A000142(n). (End)

A059565 Beatty sequence for e^gamma (gamma is the Euler-Mascheroni constant A001620).

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 14, 16, 17, 19, 21, 23, 24, 26, 28, 30, 32, 33, 35, 37, 39, 40, 42, 44, 46, 48, 49, 51, 53, 55, 56, 58, 60, 62, 64, 65, 67, 69, 71, 73, 74, 76, 78, 80, 81, 83, 85, 87, 89, 90, 92, 94, 96, 97, 99, 101, 103, 105, 106, 108, 110, 112, 113, 115, 117

Offset: 1

Mitch Harris, Jan 22 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt, Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Cf. A073004. Beatty complement is A059566.

R:=RealField(100); [Floor(Exp(EulerGamma(R))*n): n in [1..100]]; // G. C. Greubel, Aug 27 2018

Table[ Floor[ n * E^EulerGamma], {n, 1, 70} ]

{ default(realprecision, 100); b=exp(1)^Euler; for (n = 1, 2000, write("b059565.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

A379367 Numerators of the partial sums of the reciprocals of the squarefree kernel function (A007947).

Original entry on oeis.org

1, 3, 11, 7, 38, 27, 199, 117, 386, 793, 8933, 1553, 20574, 41863, 127591, 71303, 1227166, 2539417, 48759433, 24864701, 25095646, 50632187, 1174239991, 605711068, 125604071, 252924241, 267797099, 19356010, 564511331, 1891973791, 58959268151, 31867258958, 8730535499

Offset: 1

Amiram Eldar, Dec 21 2024

			Fractions begin with 1, 3/2, 11/6, 7/3, 38/15, 27/10, 199/70, 117/35, 386/105, 793/210, 8933/2310, 1553/385, ...

Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions, North-Holland Publishing Company, Amsterdam, Netherlands, 1980. See pp. 16-17.

Amiram Eldar, Table of n, a(n) for n = 1..1000
N. G. de Bruijn, On the number of integers <= x whose prime factors divide n, Illinois Journal of Mathematics, Vol. 6, No. 1 (1962), pp. 137-141.
Olivier Robert and Gérald Tenenbaum, Sur la répartition du noyau d'un entier, Indagationes Mathematicae, Vol. 24, No. 4 (2013), pp. 802-914.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.6, pp. 24-26.

Cf. A007947, A073355, A370896, A379368 (denominators), A379369.

Cf. A059956, A073004.

rad[n_] := Times @@ FactorInteger[n][[;;, 1]]; Numerator[Accumulate[Table[1/rad[n], {n, 1, 50}]]]

rad(n) = vecprod(factor(n)[, 1]);
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / rad(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} 1/A007947(k)).
a(n)/A379368(n) = exp((1 + o(1)) * sqrt(8*log(n)/log(log(n)))).
a(n)/A379368(n) ~ (1/2) * exp(gamma) * F(log(n)) * log(log(n)), where F(t) = (6/Pi^2) * Sum_{m>=1} min(1,exp(t)/m)/Product_{primes p|m} (p+1).

A119806 Decimal expansion of cos(gamma).

Original entry on oeis.org

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

Offset: 0

T. D. Noe, May 24 2006

This is the real part of exp(i*gamma), where gamma is the Euler-Mascheroni constant A001620. See A119807 for the imaginary part. The constant exp(gamma) (A073004) appears in many formulas. Does exp(i*gamma)?

			0.8379852878801965399549928612589497248086592013241766579...

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. M. Bătineţu-Giurgiu and Neculai Stanciu, Problem UP.328, Romanian Mathematical Magazine, Vol. 30, Autumn edition (2021), p. 114; Solutions, by Mokhtar Khassani-Mostaganem and Marian Ursărescu.
D. M. Bătineţu-Giurgiu, Neculai Stanciu, and José Luis Díaz-Barrero, The Last Three Decades of Lalescu Limit, Arhimede Mathematical Journal, Vol. 7, No. 1 (2020), pp. 16-26. See Problem 7, pp. 23-24.
Toyesh Prakash Sharma, The Applications of the Stirling's approximation to find limits, Revista Electronica MateInfo.ro, December 2020, pp. 44-49. See Problem 4, p. 46.

Cf. A001620 (Euler-Mascheroni constant), A073004, A119807.

SetDefaultRealField(RealField(100)); R:= RealField(); Cos(EulerGamma(R)); // G. C. Greubel, Aug 30 2018

Mathematica
```
RealDigits[Cos[EulerGamma],10,150][[1]]
```

default(realprecision, 100); cos(Euler) \\ G. C. Greubel, Aug 30 2018

Equals 2 * e * lim_{n->oo} (sin(gamma(n))-sin(gamma))*(n!)^(1/n), where gamma(n) = Sum_{k=1..n} 1/k - log(n) (Bătineţu-Giurgiu, 2021). - Amiram Eldar, Apr 02 2022

A119807 Decimal expansion of sin(gamma).

Original entry on oeis.org

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

Offset: 0

T. D. Noe, May 24 2006

This is the imaginary part of exp(i*gamma), where gamma is the Euler-Mascheroni constant A001620. See A119806 for the real part. The constant exp(gamma) (A073004) appears in many formulas. Does exp(i*gamma)?

			0.54569282320399278815735650016143074350378810920522...

G. C. Greubel, Table of n, a(n) for n = 0..10000

SetDefaultRealField(RealField(100)); R:= RealField(); Sin(EulerGamma(R)); // G. C. Greubel, Aug 30 2018

Mathematica
```
RealDigits[Sin[EulerGamma],10,150][[1]]
```

default(realprecision, 100); sin(Euler) \\ G. C. Greubel, Aug 30 2018

cp's OEIS Frontend

A002389 Decimal expansion of -log(gamma), where gamma is Euler's constant A001620.

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A094644 Continued fraction for e^gamma.

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A236435 Numerator of Product_{k=1..n-1} (1 + 1/prime(k)).

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A052488 a(n) = floor(n*H(n)) where H(n) is the n-th harmonic number, Sum_{k=1..n} 1/k (A001008/A002805).

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A091724 Decimal expansion of e^(2*EulerGamma).

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A227242 Decimal expansion of (e^gamma - 1)/e^gamma.

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