A087197 -id:A087197 - cp's OEIS frontend

A088540 Decimal expansion of (4/sqrt(Pi))exp(-gamma/2)K where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni constant.

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

Offset: 1

Benoit Cloitre, Nov 16 2003

An illustration of the Chebyshev effect.

			1.2923041571286886071...

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 100.

Gareth A. Jones and Alexander K. Zvonkin, A number-theoretic problem concerning pseudo-real Riemann surfaces, arXiv:2401.00270 [math.NT], 2023. See page 5.
S. Uchiyama, On some products involving primes, Proc. Amer. Math. Soc. 28 (1971) 629-630; MR 43#3227.

Cf. A087197, A064533, A088541, A155739.

digits = 105; LandauRamanujanK = 1/Sqrt[2]*NProduct[((1 - 2^(-2^n))*Zeta[2^n]/DirichletBeta[2^n])^(1/2^(n + 1)), {n, 1, 24}, WorkingPrecision -> digits + 5]; 4/Sqrt[Pi]*Exp[-EulerGamma/2]*LandauRamanujanK // RealDigits[#, 10, digits] & // First (* Jean-François Alcover, Jun 04 2014, updated Mar 14 2018 *)

Equals (4/sqrt(Pi))*exp(-gamma/2)*K = lim_{x->oo} Product_{p prime, p == 1 (mod 4), p <= x} (1 - 1/p).

Equals 4*A087197*A064533/exp(A155739). - R. J. Mathar, Feb 05 2009

Offset corrected by R. J. Mathar, Feb 05 2009

A222391 Decimal expansion of e^2/sqrt(Pi).

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Mar 19 2013

			4.1688284832666922304213039077523102603866646811484996378300089546240432...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A072334, A087197, A222392.
Cf. A096789: Sum_{n >= 1} 1/(Gamma(n)*Gamma(n+1)).
Cf. A035009 (see fourth comment).

Digits:=100: evalf(exp(1)^2/sqrt(Pi)); # Wesley Ivan Hurt, Jan 09 2017

Mathematica
```
RealDigits[E^2/Sqrt[Pi], 10, 90][[1]]
```

(exp(1))^2/sqrt(Pi) \\ G. C. Greubel, Jan 09 2017

Equals A072334*A087197.

Equals decimal expansion of Sum_{n >= 1} 1/(Gamma(n/2)*Gamma((n+1)/2)).

A249521 Decimal expansion of 4/sqrt(Pi), the average distance between two random Gaussian points in three dimensions.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 31 2014

Coordinates are independent normally distributed random variables with mean 0 and variance 1.

			2.25675833419102514779231780624309034337620251731...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Steven R. Finch, Random Triangles, Jan 21 2010. [Cached copy, with permission of the author]
Index entries for transcendental numbers.

Cf. A002161 (the analog constant in two dimensions), A087197, A190732 (the analog constant in one dimension).

RealDigits[4/Sqrt[Pi], 10, 103] // First

4/sqrt(Pi) \\ G. C. Greubel, Jan 09 2017

Equals Sum_{k>=0} k!/(k+3/2)!. - Amiram Eldar, Jun 19 2023

A383278 The number of integers k such that A034444(k) * k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 24, 24, 24, 24, 24, 24

Offset: 1

Amiram Eldar, Apr 21 2025

The number of terms of A383276 not exceeding n.

József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, section 31, page 72.

Amiram Eldar, Table of n, a(n) for n = 1..10000
H. L. Abbott and M. V. Subbarao, On the Distribution of the Sequence {nd*(n)}, Canadian Mathematical Bulletin , Vol. 32 , No. 1 (1989), pp. 105-108.

Partial sums of A383277.
The unitary analog of A356005.
Cf. A034444, A087197, A345288, A383276, A383279.

Accumulate[Table[DivisorSum[n, 1 &, # * 2^PrimeNu[#] == n &], {n, 1, 100}]]
(* second program: *)
f[n_] := Module[{e = IntegerExponent[n, 2], w}, w = PrimeNu[n/2^e]; If[e > w + 1 || e == w, 1, 0]]; Accumulate[Array[f, 100]]

list(lim) = my(s = 0); for(n = 1, lim, s += sumdiv(n, d, (1 << omega(d)) * d == n); print1(s, ", "));

f(n) = {my(e = valuation(n, 2), w = omega(n >> e)); e > w + 1 || e == w;}
list(lim) = my(s = 0); for(n = 1, lim, s += f(n); print1(s, ", "));

a(n) = Sum_{k=1..n} A383277(k).

a(n) = (c + o(1)) * n / sqrt(log(n)), where c = (1/sqrt(Pi)) * Product_{p prime} (p-1/2)/sqrt(p*(p-1)) = A087197 * A345288 = 0.61890644913204789046... (Abbott and Subbarao, 1989).

A087199 Decimal expansion of (3/(4*Pi))^(1/3).

Original entry on oeis.org

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

Offset: 0

Sven Simon, Aug 24 2003

Radius of a sphere of volume 1.

			0.62035049089940001666800681204777816735078620018600162009895688991314690606003....

Index entries for transcendental numbers

Cf. A086201, A087197, A087198.

RealDigits[(3/(4 Pi))^(1/3), 10, 111][[1]] (* Robert G. Wilson v, Sep 29 2014 *)

A243454 Decimal expansion of the variance of the maximum of a size 5 sample from a normal (0,1) distribution.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 05 2014

			0.44753406902066198876568465773...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

Cf. A087197, A188340, A243446, A243447, A243448, A243452, A243453.

RealDigits[1 - 25/Pi + 150*ArcCsc[Sqrt[3]]/Pi^2 + 5*Sqrt[3]*ArcSec[2*Sqrt[2/3]]/Pi^2 - 225*ArcCsc[Sqrt[3]]^2/Pi^3, 10, 101] // First

1 - 25/Pi + 150*arccsc(sqrt(3))/Pi^2 + 5*sqrt(3)*arcsec(2*sqrt(2/3))/Pi^2 - 225*arccsc(sqrt(3))^2/Pi^3.

A243523 Decimal expansion of the expectation of the maximum of a size 6 sample from a normal (0,1) distribution.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 06 2014

			1.26720636061147129764934888186399444269365...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

Cf. A087197, A243446, A243448, A243453.

digits=100; mu[6] = (15*(Pi*(Pi - 4*ArcCsc[Sqrt[3]]) + 2*NIntegrate[ ArcSin[Sqrt[3]*Sqrt[1/(8 - Tan[x]^2)]], {x, 0, ArcCsc[Sqrt[3]]}, WorkingPrecision -> digits + 5]))/(2* Pi^(5/2)) ; RealDigits[mu[6], 10, digits] // First

(15*(Pi*(Pi-4*arccsc(sqrt(3))) + 2*integral_(x=0..arccsc(sqrt(3)))(arcsin(sqrt(3)*sqrt(1/(8-tan(x)^2))))))/(2*Pi^(5/2)).

A093604 Decimal expansion of D/2, where D^2 = 3*sqrt(3)/Pi.

Original entry on oeis.org

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

Offset: 0

Lekraj Beedassy, May 14 2004

D/2=sqrt(3*sqrt(3)/Pi)/2 corresponds to the radius of the area-bisecting concentric circle within the unit-sided hexagon.

			sqrt(3*sqrt(3)/Pi)/2 = 0.6430370685787437846417825056651579788623049833326304871239...

Index entries for transcendental numbers

Cf. A097603, A010527, A011002, A087197. - R. J. Mathar, Feb 06 2009

RealDigits[Sqrt[(3Sqrt[3])/Pi]/2,10,120][[1]] (* Harvey P. Dale, Aug 27 2017 *)

sqrt(sqrt(27)/Pi)/2 \\ Charles R Greathouse IV, Oct 01 2022

Removed leading zero and adjusted offset - R. J. Mathar, Feb 06 2009

Corrected and extended by Harvey P. Dale, Aug 27 2017

A243524 Decimal expansion of the expectation of the maximum of a size 7 sample from a normal (0,1) distribution.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 06 2014

			1.3521783756069043992289221668285773452932858435...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

Cf. A087197, A243446, A243448, A243453, A243523.

digits = 100; mu[7] = (21*(Pi*(Pi - 5*ArcCsc[Sqrt[3]]) + 5*NIntegrate[ ArcSin[Sqrt[3]*Sqrt[1/(8 - Tan[x]^2)]], {x, 0, ArcCsc[Sqrt[3]]}, WorkingPrecision -> digits + 5]))/(2* Pi^(5/2)); RealDigits[mu[7], 10, digits] // First

(21*(Pi*(Pi-5*arccsc(sqrt(3))) + 5*integral_(x=0..arccsc(sqrt(3)))(arcsin(sqrt(3)*sqrt(1/(8-tan(x)^2))))))/(2*Pi^(5/2)).

A351400 Decimal expansion of e * erf(1), where erf is the error function.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 10 2022

The sum of reciprocals of the factorials of the positive half-integers.

			2.29069825230323823094953712686214731693708759053570...

Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, and Sergei Rogosin, Mittag-Leffler Functions, Related Topics and Applications, New York, NY: Springer, 2020. See p. 94, eq. (4.12.9.5).
Constantin Milici, Gheorghe Drăgănescu, and J. Tenreiro Machado, Fractional Differential Equations, Introduction to Fractional Differential Equations, Springer, Cham, 2019. See p. 12, eq. (1.9).

Eric Weisstein's World of Mathematics, Erf.
Eric Weisstein's World of Mathematics, Mittag-Leffler Function.
Wikipedia, Mittag-Leffler function.

Cf. A000079, A001113, A001147, A087197, A099286, A125961, A351401.

evalf(exp(1)*erf(1), 120);  # Alois P. Heinz, Feb 10 2022

Mathematica
```
RealDigits[E * Erf[1], 10, 100][[1]]
```

exp(1)*(1 - erfc(1)) \\ Michel Marcus, Feb 10 2022

Equals Sum_{k>=0} 1/(k + 1/2)! = Sum_{k>=1} 1/Gamma(k + 1/2).
Equals E_{1, 3/2}(1), where E_{a,b}(z) is the two-parameter Mittag-Leffler function.
Equals (1/sqrt(Pi)) * Sum_{k>=1} 2^k/(2*k-1)!! = (1/sqrt(Pi)) * Sum_{k>=1} A000079(k)/A001147(k).
Equals A001113 * A099286.
Equals A087197 * A125961.

cp's OEIS Frontend

A088540 Decimal expansion of (4/sqrt(Pi))*exp(-gamma/2)*K where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni constant.

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A222391 Decimal expansion of e^2/sqrt(Pi).

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A249521 Decimal expansion of 4/sqrt(Pi), the average distance between two random Gaussian points in three dimensions.

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A383278 The number of integers k such that A034444(k) * k <= n.

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A087199 Decimal expansion of (3/(4*Pi))^(1/3).

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A243454 Decimal expansion of the variance of the maximum of a size 5 sample from a normal (0,1) distribution.

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A243523 Decimal expansion of the expectation of the maximum of a size 6 sample from a normal (0,1) distribution.

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Mathematica

A088540 Decimal expansion of (4/sqrt(Pi))exp(-gamma/2)K where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni constant.