A064619 -id:A064619 - cp's OEIS frontend

A069998 Decimal expansion of sqrt(Pi/2).

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

Offset: 1

Benoit Cloitre, May 01 2002

This constant, sqrt(Pi/2), appears in one of the formulations of the Birthday Problem: An asymptotic expansion of the expected value for the average number of people required to find a pair having the same birthday out of k possible birthdays is sqrt(Pi/2)*sqrt(k) + 2/3 + 1/12*sqrt(Pi/2)*sqrt(1/k) - 4/135*1/k + ... found by the Indian mathematician Srinivasa Ramanujan (1887-1920). - Martin Renner, Sep 14 2016

			Sqrt(Pi/2) = 1.253314137315500251207882642... - _Wesley Ivan Hurt_, Sep 22 2016

G. C. Greubel, Table of n, a(n) for n = 1..5000
P. Flajolet, P. J. Grabner, P. Kirschenhofer and H. Prodinger, On Ramanujan's Q-Function, Journal of Computational and Applied Mathematics 58 (1995), 103-116.
I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products, (1980), page 420 (formulas 3.757.1, 3.757.2).
MIT Integration Bee, 2024 MIT Integration Bee - Semifinals, Problem 3.
Index entries for transcendental numbers

Cf. A064619.

Digits:=100: evalf(sqrt(Pi/2)); # Wesley Ivan Hurt, Sep 22 2016

RealDigits[Sqrt[Pi/2], 10, 120][[1]] (* Harvey P. Dale, Jul 24 2012 *)

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

intnum(x=0, [oo, -I], sin(x)/sqrt(x)) \\ Gheorghe Coserea, Sep 23 2018

intnum(x=[0, -1/2], [oo, I], cos(x)/sqrt(x)) \\ Gheorghe Coserea, Sep 23 2018

From A.H.M. Smeets, Sep 22 2018: (Start)
Equals Integral_{x >= 0} sin(x)/sqrt(x) dx [see Gradsteyn and Ryzhik].
Equals Integral_{x >= 0} cos(x)/sqrt(x) dx [see Gradsteyn and Ryzhik]. (End)
Equals Integral_{x>=0} (sin(x)-x*cos(x))/x^(3/2) dx. - Amiram Eldar, May 08 2021
Equals Integral_{x=0..Pi/2} sin(cot(x)^2)*sec(x)^2 dx. See MIT link. - Kritsada Moomuang, Jun 23 2025

A072829 Greatest m such that Product_{k=1..n-1} (1 - k/m) <= 1/2.

Original entry on oeis.org

2, 5, 9, 16, 23, 32, 42, 54, 68, 82, 99, 116, 135, 156, 178, 201, 226, 252, 280, 309, 340, 372, 406, 441, 477, 515, 554, 595, 637, 681, 726, 772, 820, 869, 920, 973, 1026, 1081, 1138, 1196, 1256, 1316, 1379, 1443, 1508, 1575, 1643, 1712, 1783, 1856, 1930, 2005

Offset: 2

Lekraj Beedassy, Jul 22 2002

nonn

Among n randomly selected dates over an interval of m days (or less), the odds are even (or better than even) for two or more of them to coincide.

			Thus a(7)=32 for instance implies that among 7 persons bearing the same astrological sign(extending over 30 days or so) the odds are trifle better than even for at least two of them further sharing a common birthday.

Cf. A033810, A064619.

f[n_] := (k = 1; While[ Product[1 - i/k, {i, 1, (n - 1)}] <= 1/2, k++ ]; Return[k - 1]); Table[ f[n], {n, 2, 53}]

from math import factorial, comb
def A072829(n):
    f = factorial(n)
    def p(m): return comb(m,n)*f<<1
    kmin, kmax = n-1, n
    while p(kmax) <= kmax**n: kmax<<=1
    while kmax-kmin > 1:
        kmid = kmax+kmin>>1
        if p(kmid) > kmid**n:
            kmax = kmid
        else:
            kmin = kmid
    return kmin # Chai Wah Wu, Jan 21 2025

Corresponds to the ultimate occurrence of n in A033810. For large n, m has magnitude n^2 / 2 * log(2).

Edited and extended by Robert G. Wilson v, Jul 23 2002

More terms from David Terr, Jan 03 2005

A051008 Continued fraction expansion of sqrt(2*log(2)).

Original entry on oeis.org

1, 5, 1, 1, 1, 3, 27, 1, 1, 1, 1, 24, 1, 3, 1, 24, 1, 6, 1, 8, 1, 11, 1, 3, 1079, 1, 3, 1, 1, 6, 1, 3, 10, 167, 1, 5, 1, 1, 2, 4, 6, 2, 1, 3, 2, 1, 2, 2, 9, 4, 5, 1, 20, 1, 3, 1, 2, 1, 19, 12, 4, 2, 1, 2, 4, 46, 2, 3, 20, 2, 1, 1, 2, 1, 5, 1, 1, 1, 1, 1, 1, 2, 3, 2, 6, 1, 2, 1, 13, 12, 4, 2, 1, 2, 19, 2, 1

Offset: 0

N. J. A. Sloane, Jun 07 2002

			1.177410022515474691011569326... = 1 + 1/(5 + 1/(1 + 1/(1 + 1/(1 + ...)))). - _Harry J. Smith_, Sep 20 2009

Harry J. Smith, Table of n, a(n) for n = 0..19999
Index entries for continued fractions for constants

Cf. A064619 (decimal expansion).

SetDefaultRealField(RealField(150)); ContinuedFraction(Sqrt(2*Log(2))); // G. C. Greubel, Aug 16 2018

ContinuedFraction[Sqrt[2Log[2]],100] (* Harvey P. Dale, Aug 10 2011 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(2*log(2))); for (n=1, 20000, write("b051008.txt", n-1, " ", x[n])) } \\ Harry J. Smith, Sep 20 2009

Offset changed by Andrew Howroyd, Aug 03 2024

A071856 Factorial expansion of sqrt(2ln(2)) : sqrt(2ln(2)) = sum( n>=1, a(n)/n! ).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 5, 1, 1, 5, 5, 4, 8, 4, 9, 12, 10, 11, 11, 13, 4, 10, 2, 12, 22, 13, 8, 4, 2, 16, 22, 25, 30, 13, 12, 11, 12, 8, 11, 13, 30, 7, 34, 15, 9, 18, 41, 17, 38, 23, 21, 39, 4, 46, 45, 50, 8, 45, 57, 6, 5, 41, 62, 36, 64, 1, 26, 56, 5, 29, 13, 12, 10, 1, 55, 20, 9, 40, 36

Offset: 1

Benoit Cloitre, Jun 09 2002

nonn

a(1)=1, then PARI program gives a(n) for n>1.

Cf. A064619.

for(n=2,200,c=sqrt(2*log(2)); print1(floor(n!*c)-n*floor((n-1)!*c),","))

A071857 Engel expansion of sqrt(2*log(2)).

Original entry on oeis.org

1, 6, 16, 32, 279, 726, 4141, 4368, 54482, 112572, 366613, 978019, 5342223, 41589964, 201780051, 353794663, 408307432, 463394050, 676353989, 866725306, 999357112, 3878963429, 4169753024, 8541140255, 23422387081, 26113359872, 940995107440, 1104573841707

Offset: 1

Benoit Cloitre, Jun 09 2002

nonn

a(1)=1, then PARI program gives a(n) for n>1.

Cf. A064619, A006784.

s= sqrt(2*log(2)); for(i=1,30,s=s*ceil(1/s)-1; print1(ceil(1/s),","); );

cp's OEIS Frontend

A069998 Decimal expansion of sqrt(Pi/2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A072829 Greatest m such that Product_{k=1..n-1} (1 - k/m) <= 1/2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A051008 Continued fraction expansion of sqrt(2*log(2)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A071856 Factorial expansion of sqrt(2ln(2)) : sqrt(2ln(2)) = sum( n>=1, a(n)/n! ).

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A071857 Engel expansion of sqrt(2*log(2)).

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI