A060294 -id:A060294 - cp's OEIS frontend

A258146 Decimal expansion of (1 - 2/Pi)/2: ratio of the area of a circular segment with central angle Pi/2 and the area of the corresponding circular half-disk.

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

Offset: 0

Wolfdieter Lang, May 31 2015

The formula for the ratio of the area of a circular segment with central angle alpha and the area of one half of the corresponding circular disk is (alpha - sin(alpha))/Pi. Here alpha = Pi/2.
This is also the ratio of the area of a circular disk without a central inscribed rectangle (2*x, 2*y) together with the two opposite circular segments each with central angle beta and the area of the circular disk. This is the analog of the ratio of the volume of a sphere with missing central cylinder symmetric hole of length 2*y and the area of the sphere. See a comment on A019699. In two dimensions this problem is not remarkable, because the radius R of the circle does matter. The formula is here: area ratio ar = 1 - (beta + sin(beta)/Pi) where beta = arcsin(2*yhat*sqrt(1-yhat^2)), with yhat = y/R, and beta = Pi - alpha from above.
The astonishing result from three dimensions, ar_3 = yhat^3, could suggest ar = yhat^2, which is wrong. Thanks to Sven Heinemeyer for inspiring me to look into this.
Essentially the same digit sequence as A188340. - R. J. Mathar, Jun 12 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Circular Segment.
Index entries for transcendental numbers

Cf. A019699, A060294, A258147, A258148.

RealDigits[(1-2/Pi)/2,10,120][[1]] (* Harvey P. Dale, Sep 23 2017 *)

1/2 - 1/Pi \\ Michel Marcus, Oct 19 2017

Area ratio ar = (1 - 2/Pi)/2 = 0.181690113816209...

For Buffon's constant 2/Pi see A060294.

A082542 a(n) = prime(n) + 2 - (prime(n) mod 4).

Original entry on oeis.org

2, 2, 6, 6, 10, 14, 18, 18, 22, 30, 30, 38, 42, 42, 46, 54, 58, 62, 66, 70, 74, 78, 82, 90, 98, 102, 102, 106, 110, 114, 126, 130, 138, 138, 150, 150, 158, 162, 166, 174, 178, 182, 190, 194, 198, 198, 210, 222, 226, 230, 234, 238, 242, 250, 258, 262, 270, 270, 278

Offset: 1

Reinhard Zumkeller, May 02 2003

For k > 1: a(k+1) = a(k) if and only if prime(k) == 1 modulo 4 and prime(k+1) = prime(k) + 2, see A071695 and A071696.

			a(2) = 2 because the second prime is 3, and 3 + 2 - 3 = 2.
a(3) = 6 because the third prime is 5, and 5 + 2 - 1 = 6.
a(4) = 6 because the fourth prime is 7, and 7 + 2 - 3 = 6.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000040, A060294, A070750, A076342.

Cf. A039702, A071695, A071696, A100484.

[2 + NthPrime(n) - (NthPrime(n) mod 4): n in [1..60]]; // G. C. Greubel, Nov 14 2018

Table[Prime[n] + 2 - Mod[Prime[n], 4], {n, 60}] (* Alonso del Arte, Feb 23 2015 *)
#+2-Mod[#,4]&/@Prime[Range[60]] (* Harvey P. Dale, Aug 24 2025 *)

vector(60, n, 2 + prime(n) - lift(Mod(prime(n),4))) \\ G. C. Greubel, Nov 14 2018

a(n) = A000040(n) + A070750(n).
a(n+1) = p + (-1/p) = p + (-1)^((p-1)/2), where p is the n-th odd prime and (-1/p) denotes the value of Legendre symbol. - Lekraj Beedassy, Mar 17 2005
a(n) = (A000040(n) OR 3) - 1. - Jon Maiga, Nov 14 2018
From Amiram Eldar, Dec 24 2022: (Start)
a(n) = A100484(n) - A076342(n).
Product_{n>=1} a(n)/prime(n) = 2/Pi (A060294). (End)

A165954 Decimal expansion of sqrt(10 + 2sqrt(5))/(2Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Oct 04 2009

The ratio of the volume of a regular icosahedron to the volume of the circumscribed sphere (with circumradius a*sqrt(10 + 2*sqrt(5))/4 = a*A019881, where a is the icosahedron's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165952, and A165953. A063723 shows the order of these by size.

			0.6054613829125255833862652051280444903008454088014288933200935000838295683...

Eric Weisstein's World of Mathematics, Icosahedron.
Index entries for transcendental numbers.

Cf. A000796, A002163, A165922, A049541, A165952, A165953, A063723, A019881, A019694, A060294.

RealDigits[Sqrt[10+2Sqrt[5]]/(2Pi),10,120][[1]] (* Harvey P. Dale, Aug 27 2013 *)

PARI
```
sqrt(10+2*sqrt(5))/(2*Pi)
```

sqrt(10 + 2*sqrt(5))/(2*Pi) = sqrt(10 + 2*A002163)/(2*A000796) = 2*sin(2*Pi/5)/Pi = 2*sin(A019694)/A000796 = 2*sin(72 deg)/Pi = 2*A019881/A000796 = 2*A019881*A049541 = (2/Pi)*sin(72 deg) = A060294*A019881.

A289090 Decimal expansion of (E(|x|^3))^(1/3), with x being a normally distributed random variable.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jul 26 2017

The p-th root r(p) of the expected value E(|x|^p) for various distributions appears, for example, in chemical physics, where some interactions depend on high powers of interatomic distances.
When x is distributed normally with zero mean and standard deviation 1, r(p) evaluates to r(p) = ((p-1)!!*w(p))^(1/p), where w(p) = 1 for even p and sqrt(2/Pi) for odd p. Note that, by definition, r(2) = 1 and r(1) = w(1) = A076668.
The present constant is a = r(3).

			1.16857525496246554867047601109768527106052404816790797238351628742...

Stanislav Sykora, Table of n, a(n) for n = 1..1000
Wikipedia, Normal distribution

Cf. A060294, A076668 (p=1), A011002 (p=4), A289091 (p=5), A011350 (p=6).

ExpectedValue[Abs[#]^3&, NormalDistribution[0, 1]]^(1/3) // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Jul 28 2017 *)

\\ General code, for any p > 0:
r(p) = (sqrt(2/Pi)^(p%2)*prod(k=0,(p-2)\2,p-1-2*k))^(1/p);
a = r(3) \\ Present instance

Equals (2!!*sqrt(2/Pi))^(1/3) = (2*A076668)^(1/3).

A289091 Decimal expansion of (E(|x|^5))^(1/5), with x being a normally distributed random variable.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jul 26 2017

The 5th root r(5) of the expected value E(|x|^5) for a normal distribution with zero mean and standard deviation 1. See A289090 for more details.

			1.44879190154930528525354659881281058821340103935196780729503058015...

Stanislav Sykora, Table of n, a(n) for n = 1..1000
Wikipedia, Normal distribution

Cf. A060294, A076668 (p=1), A289090 (p=3), A011002 (p=4), A011350 (p=6).

// General code, for any p > 0:
r(p) = (sqrt(2/Pi)^(p%2)*prod(k=0,(p-2)\2,p-1-2*k))^(1/p);
a = r(5) // Present instance

a = r(5), where r(p) = ((p-1)!!*sqrt(2/Pi))^(1/p).

a = (8*A076668)^(1/5).

A132698 Decimal expansion of 8/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 26 2007

			2.5464790894703253723021402139602297925513543318473031799626775049423487621476....

Index entries for transcendental numbers

Cf. A049541, A060294, A089491, A088538, A086201.

Digits:=100; evalf(8/Pi); # Wesley Ivan Hurt, Mar 26 2014

RealDigits[N[8/Pi,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

8/Pi \\ Charles R Greathouse IV, Oct 01 2022

More terms from Vladimir Joseph Stephan Orlovsky, Dec 02 2009

A132714 Decimal expansion of 24/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			=7.639437268410976116906420641880689377654062995541909539888032514827...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Cf. A049541, A060294, A089491, A088538, A086201, A132696 to A132699, A132701, A132702.

RealDigits[N[24/Pi,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 03 2009 *)

PARI
```
24/Pi \\ G. C. Greubel, Feb 20 2017
```

24/Pi = Sum_{k>=0} ( (30*k+7)*C(2*k,k)^2*(Hypergeometric2F1[1/2 - k/2, -k/2, 1, 64])/(-256)^k ). - Alexander R. Povolotsky, Dec 20 2012

Another version of this identity is: Sum[(30*k+7) * Binomial[2k,k]^2 * (Sum[Binomial[k-m,m] * Binomial[k,m] * 16^m, {m,0,k/2}])/(256)^k, {k,0,infinity}]. - Alexander R. Povolotsky, Jan 25 2013

More terms from Vladimir Joseph Stephan Orlovsky, Dec 03 2009

A277235 Decimal expansion of 2/(Gamma(3/4))^4.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Nov 13 2016

This is the value of one of Ramanujan's series: 1 - 5*(1/2)^5 + 9*(1*3/(2*4))^5 -13*(1*3*5/(2*4*6))^5 + - ... . See the Hardy reference p.7. eq. (1.4) and pp. 105-106. For the partial sums see A278140.

The proof of Hardy and Whipple mentioned in the Hardy reference reduces this series to (2/Pi)*Morley's series (for m=1/2). For this series see A277232 and A091670.

			2/Gamma(3/4)^4 = 0.88694116857811540541152...

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publ., Providence, RI, 2002, pp. 7, 105-106, 111.

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

Cf. A014549, A060294, A074799, A074800, A088538, A091670, A263809, A277232, A278140.

SetDefaultRealField(RealField(100)); 2/(Gamma(3/4))^4; // G. C. Greubel, Oct 26 2018

RealDigits[2/(Gamma[3/4])^4, 10, 100][[1]] (* G. C. Greubel, Oct 26 2018 *)

2/gamma(3/4)^4 \\ Michel Marcus, Nov 13 2016

Equals Sum_{k=0..n} (1+4*k)*(binomial(-1/2,k))^5 = Sum_{k=0..n} (-1)^k*(1+4*k)*((2*k-1)!!/(2*k)!!)^5. The double factorials are given in A001147 and A000165 with (-1)!! := 1.
Equals A060294 * A091670.
For (1+4*k)*((2*k-1)!!/(2*k)!!)^5 see A074799(k) / A074800(k).
From Amiram Eldar, Jul 13 2023: (Start)
Equals (Gamma(1/4)/Pi)^4/2.
Equals A088538 * A014549^2.
Equals A263809/Pi. (End)

A301862 Decimal expansion of the probability of intersection of 2 random chords in a circle, where each chord is selected by a random point within the circle and a random direction.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 28 2018

			0.58663629243917776194303199135765243059423027001394...

Herbert Solomon, Geometric Probability, SIAM, 1978, p. 152.

Cf. A060294, A301863, A301864, A301865.

RealDigits[1/3 + 5/(2*Pi^2), 10, 100][[1]]

1/3 + 5/(2*Pi^2) \\ Altug Alkan, Mar 28 2018

Equals 1/3 + 5/(2*Pi^2).

A000095 Number of fixed points of GAMMA_0 (n) of type i.

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane

			G.f. = x + 2*x^2 + 2*x^5 + 4*x^10 + 2*x^13 + 2*x^17 + 2*x^25 + 4*x^26 + 2*x^29 + ...

Bruno Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 101.
Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (2).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A027748, A060294, A124010, A000089.

a000095 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f 2 e = if e == 1 then 2 else 0
   f p _ = if p `mod` 4 == 1 then 2 else 0
-- Reinhard Zumkeller, Mar 24 2012

A000095 := proc(n) local b,d: if irem(n,4) = 0 then RETURN(0); else b := 1; for d from 2 to n do if irem(n,d) = 0 and isprime(d) then b := b*(1+legendre(-1,d)); fi; od; RETURN(b); fi: end;

A000095[ 1 ] = 1; A000095[ n_Integer ] := If[ Mod[ n, 4 ]==0, 0, Fold[ #1*(1+JacobiSymbol[ -1, #2 ])&, If[ EvenQ[ n ], 2, 1 ], Select[ First[ Transpose[ FactorInteger[ n ] ] ], OddQ ] ] ]
a[ n_] := If[ n < 1, 0, Times @@ (Which[# == 1, 1, # == 2, 2 Boole[#2 == 1], Mod[#, 4] == 1, 2, True, 0] & @@@ FactorInteger[n])]; (* Michael Somos, Nov 15 2015 *)

{a(n) = my(t); if( n<=1 || n%4==0, n==1, t=1; fordiv(n, d, if( isprime(d), t *= (1 + kronecker(-1, d)))); t)}; /* Michael Somos, Jul 15 2004 */

A000095(n)=n%3 && n%4 && n%7 && n%11 && return(prod(k=1,#n=factor(n)[,1],1+kronecker(-1,n[k]))) /* the n%4 is needed, the others only reduce execution time by 34% */ \\ M. F. Hasler, Mar 24 2012

from sympy import primefactors
def A000095(n): return 0 if n%4==0 or (f:=primefactors(n)) and any(p%4==3 for p in f) else 2**len(f) # David Radcliffe, Aug 20 2025

a(n) is multiplicative with a(2) = 2, a(2^e) = 0 if e>1, a(p^e) = 2 if p == 1 mod 4 and a(p^e) = 0 if p == 3 mod 4. - Michael Somos, Jul 15 2004

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2/Pi = 0.636619... (A060294). - Amiram Eldar, Oct 15 2022

Values a(1)-a(10^4) double checked by M. F. Hasler, Mar 24 2012

cp's OEIS Frontend

A258146 Decimal expansion of (1 - 2/Pi)/2: ratio of the area of a circular segment with central angle Pi/2 and the area of the corresponding circular half-disk.

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A082542 a(n) = prime(n) + 2 - (prime(n) mod 4).

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Magma

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A165954 Decimal expansion of sqrt(10 + 2*sqrt(5))/(2*Pi).

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A289090 Decimal expansion of (E(|x|^3))^(1/3), with x being a normally distributed random variable.

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A289091 Decimal expansion of (E(|x|^5))^(1/5), with x being a normally distributed random variable.

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PARI

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A132698 Decimal expansion of 8/Pi.

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Maple

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A132714 Decimal expansion of 24/Pi.

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A165954 Decimal expansion of sqrt(10 + 2sqrt(5))/(2Pi).