A073004 -id:A073004 - cp's OEIS frontend

A001620 Decimal expansion of Euler's constant (or the Euler-Mascheroni constant), gamma.

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

Offset: 0

N. J. A. Sloane

Yee (2010) computed 29844489545 decimal digits of gamma.
Decimal expansion of 0th Stieltjes constant. - Paul Muljadi, Aug 24 2010
The value of Euler's constant is close to (18/Pi^2)*Sum_{n>=0} 1/4^(2^n) = 0.5770836328... = (6/5) * A082020 * A078585. - Arkadiusz Wesolowski, Mar 27 2012

			0.577215664901532860606512090082402431042...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 3.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 259-262.
S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 28-40, 166, 365.
C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 359.
B. Gugger, Problèmes corrigés de Mathématiques posés aux concours des Ecoles Militaires, Ecole de l'Air, 1992, option MP, 1ère épreuve, Ellipses, 1993, pp. 167-184.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.3 Infinite Series, pp. 273-274.
J. Havil, Gamma: Exploring Euler's Constant, Princeton Univ. Press, 2003.
J.-M. Monier, Analyse, Exercices corrigés, 2ème année, MP, Dunod, Exercice 4.3.14, pages 371 and 387, 1997.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 166.
Joel L. Schiff, The Laplace Transform: Theory and Applications, Springer-Verlag New York, Inc. (1999). See p. 44.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 1, equation 1:7:5 at page 13.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 28.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1990.

Harry J. Smith, Table of n, a(n) for n = 0..20000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/Pi^2 and into the formal enveloping series with rational coefficients only, arXiv:1501.00740 [math.NT], 2015-2016; Journal of Number Theory (Elsevier), Volume 158, pages 365-396, 2016.
D. Bradley, Ramanujan's formula for the logarithmic derivative of the Gamma function, arXiv:math/0505125 [math.CA], 2005.
R. P. Brent, Ramanujan and Euler's constant.
R. P. Brent and F. Johansson, A bound for the error term in the Brent-McMillan algorithm, arXiv 1312.0039 [math.NA], Nov. 2013.
C. K. Caldwell, The Prime Glossary, Euler's constant.
D. Castellanos, The ubiquitous pi, Math. Mag., 61 (1988), 67-98 and 148-163.
Chao-Ping Chen, Sharp inequalities and asymptotic series of a product related to the Euler-Mascheroni constant, Journal of Number Theory, Volume 165, August 2016, Pages 314-323.
E. Chlebus, A recursive scheme for improving the original rate of convergence to the Euler-Mascheroni constant, Amer. Math. Mnthly, 118 (2011), 268-274.
M. Coffey and Jonathan Sondow, Rebuttal of Kowalenko's paper as concerns the irrationality of Euler's constant, arXiv:1202.3093 [math.NT], 2012; Acta Appl. Math., 121 (2012), 1-3.
Dave's Math Tables, Gamma Constant.
Philippe Deléham, Letter to N. J. A. Sloane, Apr 14 1997.
Thomas and Joseph Dence, A survey of Euler's constant, Math. Mag., 82 (2009), 255-265.
Pierre Dusart, Explicit estimates of some functions over primes, The Ramanujan Journal, 2016.
Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants.
X. Gourdon and P. Sebah, The Euler constant gamma.
Kalpok Guha and Sourangshu Ghosh, Measuring Abundance with Abundancy Index, (2021).
Brady Haran and Tony Padilla, The mystery of 0.577, Numberphile video, 2016.
J. C. Kluyver, Euler's constant and natural numbers, Proc. K. Ned. Akad. Wet., 27(1-2) (1924), 142-144.
D. E. Knuth, Euler's constant to 1271 places, Math. Comp. 16 1962 275-281.
Stefan Krämer, Euler's Constant γ=0.577... Its Mathematics and History.
Richard Kreckel, 116 million digits of Euler's constant (bzipped).
A. Krowne, PlanetMath.org, Euler's constant.
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, arXiv:1303.1856 [math.NT], 2013; Bull. Amer. Math. Soc., 50 (2013), 527-628.
M. Lerch, Expressions nouvelles de la constante d'Euler, S.-B. Kgl. Bohmischen Ges. Wiss., Article XLII (1897), Prague (5 pages).
Eric Naslund, Euler-Mascheroni constant expression, further simplification, MathStackExchange.
T. Papanikolaou, Plouffe's Inverter, Euler's constant to 1000000 decimals.
Michael Penn, Euler's other constant, YouTube video (2023).
S. Plouffe, using data from J. Borwein, 170000 digits of Euler or gamma constant [archived copy of a page on WorldWideSchool.org which doesn't exist any mode, cf. "Euler" link in left column].
S. Ramanujan, A series for Euler's constant, Messenger of Math., 46 (1917), 73-80.
S. Ramanujan, Question 327, J. Ind. Math. Soc.
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; 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 hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant. With an Appendix by Sergey Zlobin, arXiv:math/0211075 [math.NT], 2002-2009; Math. Slovaca 59 (2009), 1-8.
Jonathan Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), arXiv:math/0508042 [math.NT], 2005; Additive Number Theory, Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson (D. Chudnovsky and G. Chudnovsky, eds.), Springer, 2010, pp. 331-340.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006; J. Math. Anal. Appl. 332 (1) (2007), 292-314.
Jonathan Sondow and S. Zlobin, Integrals over polytopes, multiple zeta values and polylogarithms, and Euler's constant, arXiv:0705.0732 [math.NT], 2007; Math. Notes, 84 (2008), 568-583, Erratum p. 887.
Jonathan Sondow and W. Zudilin, Euler's constant, q-logarithms and formulas of Ramanujan and Gosper, arXiv:math/0304021 [math.NT], 2003; Ramanujan J. 12 (2006), 225-244.
D. W. Sweeney, On the computation of Euler's constant, Math. Comp., 17 (1963), 170-178.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant.
Wikipedia, Stieltjes constants.
A. Y. Yee, Large computations.
Index entries for sequences related to Beatty sequences.

Cf. A002852 (continued fraction).
Cf. A073004 (exp(gamma)) and A094640 ("alternating Euler constant").
Cf. A231095 (power tower using this constant).
Cf. A199332, A252898.
Denote the generalized Euler constants, also called Stieltjes constants, by Sti(n).
Sti(0) = A001620 (Euler's constant gamma) (cf. A262235/A075266),
Sti(1/2) = A301816, Sti(1) = A082633 (cf. A262382/A262383), Sti(3/2) = A301817,
Sti(2) = A086279 (cf. A262384/A262385), Sti(3) = A086280 (cf. A262386/A262387),
Sti(4) = A086281, Sti(5) = A086282, Sti(6) = A183141, Sti(7) = A183167,
Sti(8) = A183206, Sti(9) = A184853, Sti(10) = A184854.

EulerGamma(250); // G. C. Greubel, Aug 21 2018

Maple
```
Digits := 100; evalf(gamma);
```

RealDigits[ EulerGamma, 10, 105][[1]] (* Robert G. Wilson v, Nov 01 2004 *)
(1/2) N[Sum[PolyGamma[0, 1/2 + 2^k] - PolyGamma[0, 2^k], {k, 0, Infinity }], 30] (* Dimitri Papadopoulos, Nov 30 2016 *)

default(realprecision, 20080); x=Euler; d=0; for (n=0, 20000, x=(x-d)*10; d=floor(x); write("b001620.txt", n, " ", d));  \\ Harry J. Smith, Apr 15 2009

from sympy import S
def aupton(digs): return [int(d) for d in str(S.EulerGamma.n(digs+2))[2:-2]]
print(aupton(99)) # Michael S. Branicky, Nov 22 2021

Limit_{n->oo} (1 + 1/2 + ... + 1/n - log(n)) (definition).
Sum_{n>=1} (1/n - log(1 + 1/n)), since log(1 + 1/1) + ... + log(1 + 1/n) telescopes to log(n+1) and lim_{n->infinity} (log(n+1) - log(n)) = 0.
Integral_{x=0..1} -log(log(1/x)). - Robert G. Wilson v, Jan 04 2006
Integral_{x=0..1,y=0..1} (x-1)/((1-x*y)*log(x*y)). - (see Sondow 2005)
Integral_{x=0..oo} -log(x)*exp(-x). - Jean-François Alcover, Mar 22 2013
Integral_{x=0..1} (1 - exp(-x) - exp(-1/x))/x. - Jean-François Alcover, Apr 11 2013
Equals the lim_{n->oo} fractional part of zeta(1+1/n). The corresponding fractional part for x->1 from below, using n-1/n, is -(1-a(n)). The fractional part found in this way for the first derivative of Zeta as x->1 is A252898. - Richard R. Forberg, Dec 24 2014
Limit_{x->1} (Zeta(x)-1/(x-1)) from Whittaker and Watson. 1990. - Richard R. Forberg, Dec 30 2014
exp(gamma) = lim_{i->oo} exp(H(i)) - exp(H(i-1)), where H(i) = i-th Harmonic number. For a given n this converges faster than the standard definition, and two above, after taking the logarithm (e.g., 13 digits vs. 6 digits at n=3000000 or x=1+1/3000000). - Richard R. Forberg, Jan 08 2015
Limit_{n->oo} (1/2) Sum_{j>=1} Sum_{k=1..n} ((1 - 2*k + 2*n)/((-1 + k + j*n) (k + j*n))). - Dimitri Papadopoulos, Jan 13 2016
Equals 25/27 minus lim_{x->oo} 2^(x+1)/3 - (22/27)*(4/3)^x - Zeta(Sum_{i>=1} (H_i/i^x)), letting H_i denote the i-th harmonic number. - John M. Campbell, Jan 29 2016
Limit_{x->0} -B'(x), where B(x) = -x zeta(1-x) is the "Bernoulli function". - Jean-François Alcover, May 20 2016
Sum_{k>=0} (1/2)(digamma(1/2+2^k) - digamma(2^k)) where digamma(x) = d/dx log(Gamma(x)). - Dimitri Papadopoulos, Nov 14 2016
Using the abbreviations a = log(z^2 + 1/4)/2, b = arctan(2*z) and c = cosh(Pi*z) then gamma = -Pi*Integral_{0..oo} a/c^2. The general case is for n >= 0 (which includes Euler's gamma as gamma_0) gamma_n = -(Pi/(n+1))* Integral_{0..oo} sigma(n+1)/c^2, where sigma(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n,2*k)*b^(2*k) *a^(n-2*k). - Peter Luschny, Apr 19 2018
Limit_{s->0} (Zeta'(1-s)*s - Zeta(1-s)) / (Zeta(1-s)*s). - Peter Luschny, Jun 18 2018
log(2) * (gamma - (1/2) * log(2)) = -Sum_{v >= 1} (1/2^(v+1)) * (Delta^v (log(w)/w))|{w=1}, where Delta(f(w)) = f(w) - f(w + 1) (forward difference). [This is a formula from Lerch (1897).] - _Petros Hadjicostas, Jul 21 2019
From Amiram Eldar, Jul 05 2020: (Start)
Equals Integral_{x=1..oo} (1/floor(x) - 1/x) dx.
Equals Integral_{x=0..1} (1/(1-x) + 1/log(x)) dx = Integral_{x=0..1} (1/x + 1/log(1-x)) dx.
Equals -Integral_{-oo..oo} x*exp(x-exp(x)) dx.
Equals Sum_{k>=1} (-1)^k * floor(log_2(k))/k.
Equals (-1/2) * Sum_{k>=1} (Lambda(k)-1)/k, where Lambda is the Mangoldt function. (End)
Equals Integral_{0..1} -1/LambertW(-1,-x*exp(-x)) dx = 1 + Integral_{0..1} LambertW(-1/x*exp(-1/x)) dx. - Gleb Koloskov, Jun 12 2021
Equals Sum_{k>=2} (-1)^k * zeta(k)/k. - Vaclav Kotesovec, Jun 19 2021
Equals lim_{x->oo} log(x) - Sum_{p prime <= x} log(p)/(p-1). - Amiram Eldar, Jun 29 2021
Limit_{n->oo} (2*HarmonicNumber(n) - HarmonicNumber(n^2)). After answer by Eric Naslund on Mathematics Stack Exchange, on Jun 21 2011. - Mats Granvik, Jul 19 2021
Equals Integral_{x=0..oo} ( exp(-x) * (1/(1-exp(-x)) - 1/x) ) dx (see Gugger or Monier). - Bernard Schott, Nov 21 2021
Equals 1/2 + Limit_{s->1} (Zeta(s) + Zeta(1/s))/2. - Thomas Ordowski, Jan 12 2023
Equals Sum_{j>=2} Sum_{k>=2} ((k-1)/(k*j^k)). - Mike Tryczak, Apr 06 2023
From Stefano Spezia, Oct 27 2024: (Start)
Equals Sum_{n>=1} n*(zeta(n+1) - 1)/(n + 1) [Euler] (see Finch at p. 30).
Equals lim_{n->oo} Sum_{prime p<=n} log(p/(p - 1)) - log(log(n)) (see Finch at p. 31). (End)
Equals lim_{s->1} zeta(s) - zeta(s)^2/zeta(2*s - 1)/2. - Mats Granvik, Jul 07 2025

A013661 Decimal expansion of Pi^2/6 = zeta(2) = Sum_{m>=1} 1/m^2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

"In 1736 he [Leonard Euler, 1707-1783] discovered the limit to the infinite series, Sum 1/n^2. He did it by doing some rather ingenious mathematics using trigonometric functions that proved the series summed to exactly Pi^2/6. How can this be? ... This demonstrates one of the most startling characteristics of mathematics - the interconnectedness of, seemingly, unrelated ideas." - Clawson [See Hardy and Wright, Theorems 332 and 333. - N. J. A. Sloane, Jan 20 2017]
Also dilogarithm(1). - Rick L. Shepherd, Jul 21 2004
Also Integral_{x>=0} x/(exp(x)-1) dx. [Abramowitz-Stegun, 23.2.7., for s=2, p. 807]
For the partial sums see the fractional sequence A007406/A007407.
Pi^2/6 is also the length of the circumference of a circle whose diameter equals the ratio of volume of an ellipsoid to the circumscribed cuboid. Pi^2/6 is also the length of the circumference of a circle whose diameter equals the ratio of surface area of a sphere to the circumscribed cube. - Omar E. Pol, Oct 07 2011
1 < n^2/(eulerphi(n)*sigma(n)) < zeta(2) for n > 1. - Arkadiusz Wesolowski, Sep 04 2012
Volume of a sphere inscribed in a cube of volume Pi. More generally, Pi^x/6 is the volume of an ellipsoid inscribed in a cuboid of volume Pi^(x-1). - Omar E. Pol, Feb 17 2016
Surface area of a sphere inscribed in a cube of surface area Pi. More generally, Pi^x/6 is the surface area of a sphere inscribed in a cube of surface area Pi^(x-1). - Omar E. Pol, Feb 19 2016
zeta(2)+1 is a weighted average of the integers, n > 2, using zeta(n)-1 as the weights for each n. We have: Sum_{n >= 2} (zeta(n)-1) = 1 and Sum_{n >= 2} n*(zeta(n)-1) = zeta(2)+1. - Richard R. Forberg, Jul 14 2016
zeta(2) is the expected value of sigma(n)/n. - Charlie Neder, Oct 22 2018
Graham shows that a rational number x can be expressed as a finite sum of reciprocals of distinct squares if and only if x is in [0, Pi^2/6-1) U [1, Pi^2/6). See section 4 for other results and Theorem 5 for the underlying principle. - Charles R Greathouse IV, Aug 04 2020
From Peter Bala, Aug 24 2025: (Start)
By definition, zeta(2) = lim_{n -> oo} s(n), where s(n) = Sum_{k = 1..n} 1/k^2. The convergence is slow. For example, s(50) = 1.6(25...) 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). It appears that S(n) converges to zeta(2) much more rapidly. For example, S(50) = 1.64493406684822643647241516664602(137...) gives zeta(2) correct to 32 decimal digits. Cf. A073004. (End)

			1.6449340668482264364724151666460251892189499012067984377355582293700074704032...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.
F. Aubonnet, D. Guinin and B. Joppin, Précis de Mathématiques, Analyse 2, Classes Préparatoires, Premier Cycle Universitaire, Bréal, 1990, Exercice 908, pages 82 and 91-92.
Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Perseus Books, 1996, p. 97.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 262.
W. Dunham, Euler: The Master of Us All, The Mathematical Association of America, Washington, D.C., 1999, p. xxii.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.4.1 and 5.16, pp. 20, 365.
Hardy and Wright, 'An Introduction to the Theory of Numbers'. See Theorems 332 and 333.
A. A. Markoff, Mémoire sur la transformation de séries peu convergentes en séries très convergentes, Mém. de l'Acad. Imp. Sci. de St. Pétersbourg, XXXVII, 1890.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 10, 161-162.
G. F. Simmons, Calculus Gems, Section B.15, B.24, pp. 270-271, 323-325, McGraw Hill, 1992.
Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Deutscher Verlag der Wissenschaften, Berlin, 1963, p. 99, Satz 1.
A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhäuser, Boston, 1984; see p. 261.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, page 23.

Harry J. Smith, Table of n, a(n) for n = 1..20000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
D. H. Bailey, J. M. Borwein, and D. M. Bradley, Experimental determination of Apéry-like identities for zeta(4n+2), arXiv:math/0505270 [math.NT], 2005-2006.
Peter Bala, New series for old functions.
Peter Bala, Formulas for A013661.
David Benko and John Molokach, The Basel Problem as a Rearrangement of Series, The College Mathematics Journal, Vol. 44, No. 3 (May 2013), pp. 171-176.
R. Calinger, Leonhard Euler: The First St. Petersburg Years (1727-1741), Historia Mathematica, Vol. 23, 1996, pp. 121-166.
Robin Chapman, Evaluating Zeta(2).
R. W. Clickery, Probability of two numbers being coprime.
Alessio Del Vigna, On a solution to the Basel problem based on the fundamental theorem of calculus, arXiv:2104.01710 [math.HO], 2021.
Leonhard Euler, On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.
Leonhard Euler, De summis serierum reciprocarum, E41.
R. L. Graham, On finite sums of unit fractions, Proceedings of the London Mathematical Society, s3-14 (1964), pp. 193-207. doi:10.1112/plms/s3-14.2.193
Michael D. Hirschhorn, A simple proof that zeta(2) = Pi^2/6, The Mathematical Intelligencer 33:3 (2011), pp 81-82.
Melissa Larson, Verifying and discovering BBP-type formulas, 2008.
Alain Lasjaunias and Jean-Paul Tran, A note on the equality Pi^2/6 = Sum_{n>=1} 1/n^2, arXiv:2312.02245 [math.HO], 2023.
Math. Reference Project, The Zeta Function, Zeta(2).
Math. Reference Project, The Zeta Function, Odds That Two Numbers Are Coprime".
Romeo Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012.
Jon Perry, Prime Product Paradox
Simon Plouffe, Plouffe's Inverter, Zeta(2) or Pi**2/6 to 100000 digits.
Simon Plouffe, Zeta(2) or Pi**2/6 to 10000 places.
Simon Plouffe, Zeta(2) to Zeta(4096) to 2048 digits each (gzipped file)
A. L. Robledo, value of the Riemann zeta function at s=2, PlanetMath.org.
E. Sandifer, How Euler Did It, Estimating the Basel Problem.
E. Sandifer, How Euler Did It, Basel Problem with Integrals.
C. Tooth, Pi squared over six.
Eric Weisstein's World of Mathematics, Dilogarithm.
Eric Weisstein's World of Mathematics, Riemann Zeta Function zeta(2).
Wikipedia, Basel Problem.
Wikipedia, Bailey-Borwein-Plouffe formula.
Herbert S. Wilf, Accelerated series for universal constants, by the WZ method, Discrete Mathematics & Theoretical Computer Science, Vol 3, No 4 (1999).
Index entries for transcendental numbers.
Index entries for zeta function.

Cf. A001008 (H(n): numerators), A002805 (denominators), A013679 (continued fraction), A002117 (zeta(3)), A013631 (cont.frac. for zeta(3)), A013680 (cont.frac. for zeta(4)), 1/A059956, A108625, A142995, A142999.

Cf. A000290.

pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*pi^2/6))); // Vincenzo Librandi, Oct 13 2015

evalf(Pi^2/6,120); # Muniru A Asiru, Oct 25 2018
# Calculates an approximation with n exact decimal places (small deviation
# in the last digits are possible). Goes back to ideas of A. A. Markoff 1890.
zeta2 := proc(n) local q, s, w, v, k; q := 0; s := 0; w := 1; v := 4;
for k from 2 by 2 to 7*n/2 do
    w := w*v/k;
    q := q + v;
    v := v + 8;
    s := s + 1/(w*q);
od; 12*s; evalf[n](%) end:
zeta2(1000); # Peter Luschny, Jun 10 2020

RealDigits[N[Pi^2/6, 100]][[1]]
RealDigits[Zeta[2],10,120][[1]] (* Harvey P. Dale, Jan 08 2021 *)

fpprec : 100$ ev(bfloat(zeta(2)))$ bfloat(%); /* Martin Ettl, Oct 21 2012 */

PARI
```
default(realprecision, 200); Pi^2/6
```
PARI
```
default(realprecision, 200); dilog(1)
```
PARI
```
default(realprecision, 200); zeta(2)
```

A013661(n)={localprec(n+2); Pi^2/.6\10^n%10} \\ Corrected and improved by M. F. Hasler, Apr 20 2021

default(realprecision, 20080); x=Pi^2/6; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b013661.txt", n, " ", d)); \\ Harry J. Smith, Apr 29 2009

sumnumrat(1/x^2, 1) \\ Charles R Greathouse IV, Jan 20 2022

# Use some guard digits when computing.
# BBP formula (3 / 16) P(2, 64, 6, (16, -24, -8, -6,  1, 0)).
from decimal import Decimal as dec, getcontext
def BBPzeta2(n: int) -> dec:
    getcontext().prec = n
    s = dec(0); f = dec(1); g = dec(64)
    for k in range(int(n * 0.5536546824812272) + 1):
        sixk = dec(6 * k)
        s += f * ( dec(16) / (sixk + 1) ** 2 - dec(24) / (sixk + 2) ** 2
                 - dec(8)  / (sixk + 3) ** 2 - dec(6)  / (sixk + 4) ** 2
                 + dec(1)  / (sixk + 5) ** 2 )
        f /= g
    return (s * dec(3)) / dec(16)
print(BBPzeta2(2000))  # Peter Luschny, Nov 01 2023

Limit_{n->oo} (1/n)*(Sum_{k=1..n} frac((n/k)^(1/2))) = zeta(2) and in general we have lim_{n->oo} (1/n)*(Sum_{k=1..n} frac((n/k)^(1/m))) = zeta(m), m >= 2. - Yalcin Aktar, Jul 14 2005
Equals Integral_{x=0..1} (log(x)/(x-1)) dx or Integral_{x>=1} (log(x/(x-1))/x) dx. - Jean-François Alcover, May 30 2013
For s >= 2 (including Complex), zeta(s) = Product_{n >= 1} prime(n)^s/(prime(n)^s - 1). - Fred Daniel Kline, Apr 10 2014
Also equals 1 + Sum_{n>=0} (-1)^n*StieltjesGamma(n)/n!. - Jean-François Alcover, May 07 2014
zeta(2) = Sum_{n>=1} ((floor(sqrt(n)) - floor(sqrt(n-1)))/n). - Mikael Aaltonen, Jan 10 2015
zeta(2) = Sum_{n>=1} (((sqrt(5)-1)/2/sqrt(5))^n/n^2) + Sum_{n>=1} (((sqrt(5)+1)/2/sqrt(5))^n/ n^2) + log((sqrt(5)-1)/2/sqrt(5))log((sqrt(5)+1)/2/sqrt(5)). - Seiichi Kirikami, Oct 14 2015
The above formula can also be written zeta(2) = dilog(x) + dilog(y) + log(x)*log(y) where x = (1-1/sqrt(5))/2 and y=(1+1/sqrt(5))/2. - Peter Luschny, Oct 16 2015
zeta(2) = Integral_{x>=0} 1/(1 + e^x^(1/2)) dx, because (1 - 1/2^(s-1))*Gamma[1 + s]*Zeta[s] = Integral_{x>=0} 1/(1 + e^x^(1/s)) dx. After Jean-François Alcover in A002162. - Mats Granvik, Sep 12 2016
zeta(2) = Product_{n >= 1} (144*n^4)/(144*n^4 - 40*n^2 + 1). - Fred Daniel Kline, Oct 29 2016
zeta(2) = lim_{n->oo} (1/n) * Sum_{k=1..n} A017665(k)/A017666(k). - Dimitri Papadopoulos, May 10 2019 [See the Walfisz reference, and a comment in A284648, citing also the Sándor et al. Handbook. - Wolfdieter Lang, Aug 22 2019]
Equals Sum_{k>=1} H(k)/(k*(k+1)), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Aug 16 2020
Equals (8/3)*(1/2)!^4 = (8/3)*Gamma(3/2)^4. - Gary W. Adamson, Aug 17 2021
Equals ((m+1)/m) * Integral_{x=0..1} log(Sum {k=0..m} x^k )/x dx, m > 0 (Aubonnet reference). - _Bernard Schott, Feb 11 2022
Equals 1 + Sum_{n>=1} n*(zeta(n+2)-1). - Richard R. Forberg, Jun 04 2023; improved by Natalia L. Skirrow, Jul 25 2025
Equals Psi'(1) where Psi'(x) is the trigamma function (by Abramowitz Stegun 6.4.2). - Andrea Pinos, Oct 22 2024
Equals Integral_{x=0..1} Integral_{y=0..1} 1/(1 - x*y) dy dx. - Kritsada Moomuang, May 22 2025
Equals 1 + Sum_{n>=1} 1/(n^2*(n+1)). - Natalia L. Skirrow, Jul 25 2025

Edited by N. J. A. Sloane, Nov 22 2023

A003277 Cyclic numbers: k such that k and phi(k) are relatively prime; also k such that there is just one group of order k, i.e., A000001(k) = 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, 141, 143, 145, 149, 151, 157, 159, 161, 163, 167, 173

Offset: 1

N. J. A. Sloane and Richard Stanley

Except for a(2)=2, all the terms in the sequence are odd. This is because of the existence of a non-cyclic dihedral group of order 2n for each n>1. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 09 2001
Also gcd(n, A051953(n)) = 1. - Labos Elemer
n such that x^n == 1 (mod n) has no solution 2 <= x <= n. - Benoit Cloitre, May 10 2002
There is only one group (the cyclic group of order n) whose order is n. - Gerard P. Michon, Jan 08 2008  [This is a 1947 result of Tibor Szele. - Charles R Greathouse IV, Nov 23 2011]
Any divisor of a Carmichael number (A002997) must be odd and cyclic. Conversely, G. P. Michon conjectured (c. 1980) that any odd cyclic number has at least one Carmichael multiple (if the conjecture is true, each of them has infinitely many such multiples). In 2007, Michon & Crump produced explicit Carmichael multiples of all odd cyclic numbers below 10000 (see link, cf. A253595). - Gerard P. Michon, Jan 08 2008
Numbers n such that phi(n)^phi(n) == 1 (mod n). - Michel Lagneau, Nov 18 2012
Contains A000040, and all members of A006094 except 6. - Robert Israel, Jul 08 2015
Number m such that n^n == r (mod m) is solvable for any r. - David W. Wilson, Oct 01 2015
Numbers m such that A074792(m) = m + 1. - Thomas Ordowski, Jul 16 2017
Squarefree terms of A056867 (see McCarthy link p. 592 and similar comment with "cubefree" in A051532). - Bernard Schott, Mar 24 2022

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
J. S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 7.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Max Alekseyev, Michon's conjecture (Open Problem Garden, Aug. 2007).
Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025. See pp. 1, 2, 30.
Keith Conrad, When are all groups of order n cyclic?, University of Connecticut, 2019.
Peter J. Dukes and Joanna Niezen, Pairwise balanced designs of dimension three, Australasian Journal Of Combinatorics, Volume 61(1) (2015), pages 98-113.
Paul Erdős, Some asymptotic formulas in number theory, J. Indian Math. Soc. (N.S.) 12 (1948), pp. 75-78.
Jose M. Grau, Manuel Rodríguez, A. Oller-Marcen, and Daniel Sadornil, Fermat test with gaussian base and Gaussian pseudoprimes, arXiv preprint arXiv:1401.4708 [math.NT], 2014.
Donald J. McCarthy, A survey of partial converses to Lagrange's theorem on finite groups, Transactions of the New York Academy of Sciences, Vol. 33, No. 6, Series II (1971), pp. 586-594. See page 592.
Gérard P. Michon, Carmichael Divisors
Gérard P. Michon and J. K. Crump, Carmichael Multiples of Odd Cyclic Numbers (up to 10000)
Jonathan Pakianathan and Krishnan Shankar, Nilpotent Numbers, Amer. Math. Monthly, 107, August-September 2000, 631-634.
R. P. Stanley, Letter to N. J. A. Sloane, c. 1991
T. Szele, Über die endlichen Ordnungszahlen, zu denen nur eine Gruppe gehört, Commentarii Mathematici Helvetici 20 (1947), pp. 265-267.
Index entries for sequences related to groups

Subsequence of A051532. Intersection of A056867 and A005117.
Cf. A000010, A008966, A009195, A050384 (the same sequence but with the primes removed). Also A000001(a(n)) = 1.
Cf. A002997, A006094, A054395, A055561, A054396, A054397, A056867, A135850, A249550, A249551, A249552, A249553, A249554, A249555, A036537, A051953, A253595.

import Data.List (elemIndices)
a003277 n = a003277_list !! (n-1)
a003277_list = map (+ 1) $ elemIndices 1 a009195_list
-- Reinhard Zumkeller, Feb 27 2012

[n: n in [1..200] | Gcd(n, EulerPhi(n)) eq 1]; // Vincenzo Librandi, Jul 09 2015

select(t -> igcd(t, numtheory:-phi(t))=1, [$1..1000]); # Robert Israel, Jul 08 2015

Select[Range[175], GCD[#, EulerPhi[#]] == 1 &] (* Jean-François Alcover, Apr 04 2011 *)
Select[Range@175, FiniteGroupCount@# == 1 &] (* Robert G. Wilson v, Feb 16 2017 *)
Select[Range[200],CoprimeQ[#,EulerPhi[#]]&] (* Harvey P. Dale, Apr 10 2022 *)

isA003277(n) = gcd(n,eulerphi(n))==1 \\ Michael B. Porter, Feb 21 2010

# Compare A050384.
def isPrimeTo(n, m): return gcd(n, m) == 1
def isCyclic(n): return isPrimeTo(n, euler_phi(n))
[n for n in (1..173) if isCyclic(n)] # Peter Luschny, Nov 14 2018

n = p_1*p_2*...*p_k (for some k >= 0), where the p_i are distinct primes and no p_j-1 is divisible by any p_i.
A000001(a(n)) = 1.
Erdős proved that a(n) ~ e^gamma n log log log n, where e^gamma is A073004. - Charles R Greathouse IV, Nov 23 2011
A000005(a(n)) = 2^k. - Carlos Eduardo Olivieri, Jul 07 2015
A008966(a(n)) = 1. - Bernard Schott, Mar 24 2022

More terms from Christian G. Bower

A080130 Decimal expansion of exp(-gamma).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Jan 26 2003

By Mertens's third theorem, lim_{k->oo} (H_{k-1}*Product_{prime p<=k} (1-1/p)) = exp(-gamma), where H_n is the n-th harmonic number. Let F(x) = lim_{n->oo} ((Sum_{k<=n} 1/k^x)*(Product_{prime p<=n} (1-1/p^x))) for real x in the interval 0 < x < 1. Consider the function F(s) of the complex variable s, but without the analytic continuation of the zeta function, in the critical strip 0 < Re(s) < 1. - Thomas Ordowski, Jan 26 2023

			0.56145948356688516982414321479088078676571...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.5 p. 29, 2.7 p. 117 and 5.4 p. 285.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 202.
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., 50 (2013), 527-628; arXiv:1303.1856 [math.NT], 2013.
Doron Zeilberger and Noam Zeilberger, Fractional Counting of Integer Partitions, 2018; Local copy [Pdf file only, no active links]

Cf. A000010, A000142, A001113, A001620, A002852, A007838, A073004, A079650, A322364, A322365, A322380, A322381.

R:= RealField(100); Exp(-EulerGamma(R)); // G. C. Greubel, Aug 28 2018

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

RealDigits[N[Exp[-EulerGamma], 200]][[1]] (* Arkadiusz Wesolowski, Aug 26 2012 *)

default(realprecision, 100); exp(-Euler) \\ G. C. Greubel, Aug 28 2018

Equals lim inf_{n->oo} phi(n)*log(log(n))/n. - Arkadiusz Wesolowski, Aug 26 2012
From Alois P. Heinz, Dec 05 2018: (Start)
Equals lim_{n->oo} A322364(n)/(n*A322365(n)).
Equals lim_{n->oo} A322380(n)/A322381(n). (End)
Equals lim_{k->oo} log(k)*Product_{prime p<=k} (1-1/p). - Amiram Eldar, Jul 09 2020
Equals lim_{n->oo} A007838(n)/A000142(n). - Alois P. Heinz, Feb 24 2022
Equals Product_{k>=1} (1+1/k)*exp(-1/k). - Amiram Eldar, Mar 20 2022
Equals A001113^(-A001620). - Omar E. Pol, Dec 14 2022
Equals lim_{n->oo} (A001008(p_n-1)/A002805(p_n-1))*(A038110(n+1)/A060753(n+1)), where p_n = A000040(n). - Thomas Ordowski, Jan 26 2023

A054640 a(n) is the sum of the divisors of the n-th primorial: a(n) = A000203(A002110(n)).

Original entry on oeis.org

1, 3, 12, 72, 576, 6912, 96768, 1741824, 34836480, 836075520, 25082265600, 802632499200, 30500034969600, 1281001468723200, 56364064623820800, 2705475101943398400, 146095655504943513600, 8765739330296610816000, 543475838478389870592000, 36956357016530511200256000

Offset: 0

Labos Elemer, May 15 2000

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Rafael Jakimczuk, Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts, International Mathematical Forum, Vol. 12, No. 7 (2017), pp. 331-338.

Cf. A000203, A002110, A005867, A008864, A028815, A051674, A054641, A070826, A073004, A074107.

[1/2*&*[(1+NthPrime(k)): k in [0..n-1]]: n in [1..19]]; // Vincenzo Librandi, May 08 2017

a:= n-> mul(1+ithprime(j), j=1..n): seq(a(n), n=0..20); # Zerinvary Lajos, Aug 24 2008

Table[Product[1 + Prime[i], {i,n-1}], {n,100}] (* Geoffrey Critzer, Dec 01 2014 *)

a(n)=prod(i=1,n,prime(i)+1) \\ Charles R Greathouse IV, Feb 13 2013

def A054640(n): return product(nth_prime(j)+1 for j in range(1,n+1))
[A054640(n) for n in range(41)] # G. C. Greubel, Aug 05 2024

a(n+1) = a(n)*(prime(n) + 1) = a(n)*A028815(n) (quotient=n-th prime+1 starting with 2).
a(n) ~ (6/Pi^2) * exp(gamma) * A002110(n) * log(prime(n)) + O(A002110(n)) (Jakimczuk, 2017). - Amiram Eldar, Feb 17 2021
a(n) = a(n-1) * A008864(n). - Flávio V. Fernandes, Mar 20 2021
a(n) = A002110(n) + A074107(n), a(n) <= A070826(1+n) [= A002110(1+n)/2] < A051674(n). - Antti Karttunen, Nov 19 2024

a(0)=1 prepended by Alois P. Heinz, Apr 01 2021

A062569 a(n) = sigma(n!).

Original entry on oeis.org

1, 1, 3, 12, 60, 360, 2418, 19344, 159120, 1481040, 15334088, 184009056, 2217441408, 31044179712, 442487616480, 6686252969760, 107004539285280, 1926081707135040, 34683832925921088, 693676658518421760, 13891399238731734720, 292460416142501376000

Offset: 0

Jason Earls, Jul 03 2001

			a(4) = 60, since the sum of the positive divisors of 4! is 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60. - _Timothy L. Tiffin_, Jan 22 2023

Wacław Sierpiński, Elementary Theory of Numbers, Ex. 6, p. 169, Warsaw, 1964.

Paul D. Hanna, Table of n, a(n) for n = 0..300
Rafael Jakimczuk, Two topics in number theory: sum of divisors of the factorial and a formula for primes, International Mathematical Forum, Vol. 12, No. 19 (2017), pp. 929-935. See Theorem 1.4, p. 932.
Vaclav Kotesovec, Plot of a(n)/(n!*log(n)) for n = 2..50000.

Cf. A000142, A000203, A027423, A073004.

with(numtheory):seq(sigma(n!), n=0..19); # Zerinvary Lajos, Feb 15 2008

Array[DivisorSigma[1,#! ]&,33,1] (* Vladimir Joseph Stephan Orlovsky, Nov 01 2009 *)

PARI
```
for(n=0,21,print(sigma(n!)))
```

[sigma(ZZ(n).factorial(), 1) for n in range(20)]  # Zerinvary Lajos, Jun 13 2009

a(n) = A000203(A000142(n)). - Michel Marcus, Jan 10 2015
a(p) = (p+1)*a(p-1) for p prime. - Altug Alkan, Jul 18 2016
Limit_{n->oo} a(n)/n! = oo. Proof in Sierpiński. - Bernard Schott, Feb 09 2019
a(n) ~ c * n! * log(n) * (1 + O(1/log(n))), where c =  exp(gamma) = A073004 (Jakimczuk, 2017). - Amiram Eldar, Nov 07 2020

A002852 Continued fraction for Euler's constant (or Euler-Mascheroni constant) gamma.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, 11, 3, 7, 1, 7, 1, 1, 5, 1, 49, 4, 1, 65, 1, 4, 7, 11, 1, 399, 2, 1, 3, 2, 1, 2, 1, 5, 3, 2, 1, 10, 1, 1, 1, 1, 2, 1, 1, 3, 1, 4, 1, 1, 2, 5, 1, 3, 6, 2, 1, 2, 1, 1, 1, 2, 1, 3, 16, 8, 1, 1, 2, 16, 6, 1, 2, 2, 1, 7, 2, 1, 1, 1, 3, 1, 2, 1, 2

Offset: 0

N. J. A. Sloane

The first 970258158 terms were computed by Eric W. Weisstein on Sep 21 2011 using a developmental version of Mathematica.
The first 4851382841 terms were computed by Eric W. Weisstein on Jul 22 2013 using a developmental version of Mathematica.
The first 16695279010 terms were computed by Syed Fahad on Apr 29 2021, see link.

			0.577215664901532860606512090082402431042...
0 + 1/(1 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + 1/(1 + 1/(4 + 1/(3 + 1/(13 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 3.
R. S. Lehman, A Study of Regular Continued Fractions. Report 1066, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland, Feb 1959.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
K. Y. Choong, D. E. Daykin and C. R. Rathbone, Rational approximations to pi, Math. Comp., 25 (1971), 387-392.
K. Y. Choong, D. E. Daykin and C. R. Rathbone, Regular continued fractions for pi and gamma, Math. Comp., 25 (1971), 403. [Review of their report at the University of Malaya Computer Centre, September 1970.]
Donald E. Knuth, Euler's constant to 1271 places, Math. Comp. 16 (1962), 275-281.
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, arXiv:1303.1856 [math.NT], 2003.
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., 50 (2013), 527-628.
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 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.
Syed Fahad, Euler's Constant - Simple Continued Fraction (16,695,279,010 terms)
Syed Fahad, Zuuv, Multiprecision Floating-point to Continued Fraction Cruncher
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant Continued Fraction.
Gang Xiao, Contfrac.
Index entries for continued fractions for constants

Cf. A001620, the decimal expansion, which has many more references.
See also A073004 (exp(gamma)) and A094640 ("alternating Euler constant").
Cf. A033091 (incrementally largest terms), A033092 (positions of incrementally largest terms).
Cf. A033149 (positions of first occurrence of n in the continued fraction).

ContinuedFraction(EulerGamma(100)); // Vincenzo Librandi, Oct 19 2017

Mathematica
```
ContinuedFraction[EulerGamma, 100]
```

default(realprecision, 11000); x=contfrac(Euler); for (n=0, 10000, write("b002852.txt", n, " ", x[n+1])) \\ Harry J. Smith, Apr 14 2009

More terms from Robert G. Wilson v, Dec 08 2000

A067698 Positive integers such that sigma(n) >= exp(gamma) * n * log(log(n)).

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360, 720, 840, 2520, 5040

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

Previous name was: Numbers with relatively many and large divisors.
n is in the sequence iff sigma(n) >= exp(gamma) * n * log(log(n)), where gamma = Euler-Mascheroni constant and sigma(n) = sum of divisors of n.
Robin has shown that 5040 is the last element in the sequence iff the Riemann hypothesis is true. Moreover the sequence is infinite if the Riemann hypothesis is false. Gronwall's theorem says that
  lim sup_{n -> infinity} sigma(n)/(n*log(log(n))) = exp(gamma).
a(28) > 10^(10^13.11485), if the Riemann hypothesis is false (Morrill and Platt, 2021). Briggs (2006) found the lower bound 10^(10^10). - Amiram Eldar, Jul 23 2025

			9 is in the sequence since sigma(9) = 13 > 12.6184... = exp(gamma) * 9 * log(log(9)).

Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 15, No. 2 (2006), p. 251-256.
Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), #A33.
Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, On SA, CA, and GA numbers, The Ramanujan Journal, Vol. 29 (2012), pp. 359-384; arXiv preprint, arXiv:1112.6010 [math.NT], 2011-2012.
Jeffrey C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, The American Mathematical Monthly, Vol. 109, No. 6 (2002), pp. 534-543; alternative copy; arXiv preprint, arXiv:math/0008177 [math.NT], 2000-2001.
Thomas Morrill and David John Platt, Robin's inequality for 20-free integers, Integers, Vol. 21 (2021), #A28.
Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.
Eric Weisstein's World of Mathematics, Gronwall's Theorem.
Eric Weisstein's World of Mathematics, Robin's Theorem.

Cf. A057641 (based on Lagarias's extension of Robin's result).

Cf. A000203, A004394, A073004, A091901, A189686, A196229.

with (numtheory): expgam := exp(evalf(gamma)): for i from 2 to 6000 do: a := sigma (i): b := expgam*i*evalf(ln(ln(i))): if a >= b then print (i, a, b): fi: od:

fQ[n_] := DivisorSigma[1, n] > n*Exp@ EulerGamma*Log@ Log@n; lst = {}; Do[ If[ fQ[n], AppendTo[lst, n]], {n,2,10^4}]; lst (* Robert G. Wilson v, May 16 2003 *)
Select[Range[2,5050], Exp[EulerGamma] # Log[Log[#]]-DivisorSigma[1,#]<0 &] (* Ant King, Feb 28 2013 *)

is(n)=sigma(n) >= exp(Euler) * n * log(log(n)) \\ Charles R Greathouse IV, Feb 08 2017

from sympy import divisor_sigma, EulerGamma, E, log
print([n for n in range(2, 5041) if divisor_sigma(n) >= (E**EulerGamma * n * log(log(n)))]) # Karl-Heinz Hofmann, Apr 22 2022

Edited by N. J. A. Sloane at the suggestion of Max Alekseyev, Jul 17 2007

New name from Jud McCranie, Aug 14 2017

cp's OEIS Frontend

A001620 Decimal expansion of Euler's constant (or the Euler-Mascheroni constant), gamma.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

A013661 Decimal expansion of Pi^2/6 = zeta(2) = Sum_{m>=1} 1/m^2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

PARI

PARI

PARI

PARI

PARI

Python

Formula

Extensions

A003277 Cyclic numbers: k such that k and phi(k) are relatively prime; also k such that there is just one group of order k, i.e., A000001(k) = 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A080130 Decimal expansion of exp(-gamma).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A054640 a(n) is the sum of the divisors of the n-th primorial: a(n) = A000203(A002110(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma