A000796 -id:A000796 - cp's OEIS frontend

Previous Showing 61-70 of 1164 results. Next

A197682 Decimal expansion of Pi/(2 + 2*Pi).

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

Offset: 0

Clark Kimberling, Oct 17 2011

The number Pi/(2 + 2*Pi) is the least x > 0 such that sin(x) = cos(Pi*x).
If b and c are distinct real numbers, the solutions of sin(bx) = cos(cx) are x = (k - 1/2)*Pi/(b + c), where k runs through the integers. Thus, if b > 0 and c > 0, the least solution x > 0 is Pi/(2*b + 2*c), so that this is also the least x > 0 for which sin(c*x) = cos(b*x).  Related sequences, each with a Mathematica program which includes a graph:
...
b.....c.......sequence........x
1.....2.......A019673........ x = Pi/6
1.....3.......A019678........ x = Pi/8
1.....4.......(A000796)/10... x = Pi/10
1.....Pi......A197682........ x = Pi/(2+2*Pi)
1.....2*Pi....A197683........ x = Pi/(2+4*Pi)
1.....1/Pi....A197684........ x = Pi^2/(2+2*Pi)
1.....2/Pi....A197685........ x = Pi^2/(4+2*Pi)
1.....Pi/2....A197686........ x = Pi/(2+Pi)
1.....Pi/3....A197687........ x = 3*Pi/(6+2*Pi)
1.....Pi/4....A197688........ x = 2*Pi/(4+Pi)
1.....Pi/6....A197689........ x = 3*Pi/(6+Pi)
2.....3.......(A000796)/10... x = Pi/10
2.....Pi......A197690........ x = Pi/(4+2*Pi)
2.....2*Pi....A197691........ x = Pi/(4+4*Pi)
2.....1/Pi....A197692........ x = Pi^2/(2+4*Pi)
2.....2/Pi....A197693........ x = Pi^2/(4+4*Pi)
2.....Pi/2....A197694........ x = Pi/(4+Pi)
3.....Pi......A197695........ x = Pi/(2+2*Pi)
3.....2*Pi....A197696........ x = Pi/(6+4*Pi)
3.....1/Pi....A197697........ x = Pi^2/(2+6*Pi)
3.....2/Pi....A197698........ x = Pi^2/(4+6*Pi)
3.....Pi/2....A197699........ x = Pi/(6+Pi)
1/2...Pi......A197700........ x = Pi/(1+2*Pi)
1/2...2*Pi....A197701........ x = Pi/(1+4*Pi)
1/2...1/Pi....A197724........ x = Pi^2/(2+Pi)
1/2...2/Pi....A197725........ x = Pi^2/(4+Pi)
1/2...Pi/2....A197726........ x = Pi/(1+Pi)
1/2...Pi/4....A197727........ x = 2*Pi/(2+Pi)
1/3...Pi/3....A197728........ x = 3*Pi/(2+2*Pi)
1/3...Pi/6....A197729........ x = 3*Pi/(2+Pi)
2/3...Pi/6....A197730........ x = 3*Pi/(4+Pi)
1/4...Pi......A197731........ x = 2*Pi/(1+4*Pi)
1/4...Pi/2....A197732........ x = 2*Pi/(1+2*Pi)
1/4...Pi/4....A197733........ x = 2*Pi/(1+Pi)
1/5...Pi/5....10*A197691..... x = 5*Pi/(2+2*Pi)
1/6...Pi/6....A197735........ x = 3*Pi/(1+Pi)
1/8...Pi/8....A197736........ x = 4*Pi/(1+Pi)

			0.37927349649738807267221534452244643...

Index entries for transcendental numbers

Cf. A197683.

b = 1; c = Pi;
t = x /. FindRoot[Sin[b*x] == Cos[c*x], {x, .3, .4}]
N[Pi/(2*b + 2*c), 110]
RealDigits[%]  (* A197682 *)
Simplify[Pi/(2*b + 2*c)]
Plot[{Sin[b*x], Cos[c*x]}, {x, 0, Pi}]

1/(2/Pi+2) \\ Charles R Greathouse IV, Sep 27 2022

A004602 Expansion of Pi in base 3.

Original entry on oeis.org

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

Offset: 2

N. J. A. Sloane

			10.0102110122220102110021111102212222201...

G. C. Greubel, Table of n, a(n) for n = 2..10000
Steve Pagliarulo, Stu's pi page: base 3 (23 pages).

Pi in base b: A004601 (b=2), this sequence (b=3), A004603 (b=4), A004604 (b=5), A004605 (b=6), A004606 (b=7), A006941 (b=8), A004608 (b=9), A000796 (b=10), A068436 (b=11), A068437 (b=12), A068438 (b=13), A068439 (b=14), A068440 (b=15), A062964 (b=16), A060707 (b=60), A331313 (Pi in balanced ternary).

Cf. A007514.

RealDigits[ N[ Pi, 105], 3] [[1]]
RealDigits[Pi,3,120][[1]] (* Harvey P. Dale, Jul 02 2021 *)
Table[ResourceFunction["NthDigit"][Pi, n, 3], {n, 1, 100}] (* Joan Ludevid, Jun 24 2022;easy to compute a(10000000)=1 with this function;requires Mathematica 12.0+ *)

a(n) = floor(Pi*3^(n-3)) - 3*floor(Pi*3^(n-4)), n>1. - G. C. Greubel, Mar 09 2018

A063438 a(n) = floor((n+1)Pi) - floor(nPi).

Original entry on oeis.org

3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4

Offset: 1

Jason Earls, Jul 24 2001

The arithmetic mean (1/(n+1))*Sum_{k=0..n} a(k) converges to Pi. What is effectively the same: the Cesaro limit (C1) of a(n) is Pi. - Hieronymus Fischer, Jan 31 2006

A word that is uniformly recurrent, but not morphic. - N. J. A. Sloane, Jul 14 2018

			a(6)=3 because 7*Pi = 21.99..., 6*Pi = 18.84..., so a(6) = 21 - 18;
a(7)=4 because 8*Pi = 25.13..., 7*Pi = 21.99..., so a(7) = 25 - 21.

G. H. Hardy, Divergent Series, Oxford 1979.
Zeller, K. and Beekmann, W., Theorie der Limitierungsverfahren. Springer Verlag, Berlin, 1970.

Harry J. Smith, Table of n, a(n) for n = 1..2000
Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, and Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017.

Cf. A000796, A115788, A115789, A115790, A006337.
First differences of A022844.
Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: A316829, 37: A010060.

Differences[Floor[Pi Range[120]]] (* Harvey P. Dale, Jul 02 2021 *)

j=[]; for(n=1,150,j=concat(j, floor( (n+1) * Pi) - floor(n * Pi))); j

{ default(realprecision, 50); for (n=1, 2000, write("b063438.txt", n, " ", floor((n + 1)*Pi) - floor(n*Pi)) ) } \\ Harry J. Smith, Aug 21 2009

a(n) = floor((n+1)*Pi) - floor(n*Pi) \\ Michel Marcus, Jul 15 2013

a(n) = A115790(n) + 3. - Michel Marcus, Jul 15 2013

Offset in b-file and second PARI program corrected by N. J. A. Sloane, Aug 31 2009

Entry revised by N. J. A. Sloane, Jan 07 2014

A086201 Decimal expansion of 1/(2*Pi).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jul 12 2003

If a single hump of cycloid, with arc length 8*radius (generating circle), is inside a rectangle with width=2*radius and length=2*Pi*radius, then the radius must be 1/(2*Pi) (this sequence) to have (2/Pi), A060294, as semi arc of cycloid (arc = 4/Pi = A088538) and the rectangle... length = 1, width = 1/Pi. I suppose that in 3D geometry, gliding along a cycloid, in all directions around, from a point A at the height of 1/Pi, gives Pi*point B. - Eric Desbiaux, Dec 21 2008
Radius of circle having circumference 1. - Clark Kimberling, Jan 06 2014
The number of primitive Pythagorean triangles with hypotenuse less than N is approximately N/(2*Pi), found by Lehmer, cf. Knott link. - Frank Ellermann, Mar 27 2020

			0.15915494309189533576888376337251...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.4, p. 493.

Ron Knott, 9.6 Pythagorean Triples and Pi, 2019.
Eric Weisstein's World of Mathematics, Plouffe's Constants.
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Index entries for transcendental numbers.

Cf. A000796 (Pi), A019692 (2*Pi).

RealDigits[N[1/(2 Pi), 100]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

PARI
```
1/(2*Pi) \\ Michel Marcus, Mar 28 2020
```

Link corrected by Fred Daniel Kline, Jul 29 2015

A005042 Primes formed by the initial digits of the decimal expansion of Pi.

Original entry on oeis.org

3, 31, 314159, 31415926535897932384626433832795028841

Offset: 1

N. J. A. Sloane

The next term consists of the first 16208 digits of Pi and is too large to show here (see A060421). Ed T. Prothro found this probable prime in 2001.

A naive probabilistic argument suggests that the sequence is infinite. - Michael Kleber, Jun 23 2004

M. Gardner, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. Gardner, Letter to N. J. A. Sloane, Nov 16 1979.
Ed T. Prothro, How I Found the Next Pi Prime.
Eric Weisstein's World of Mathematics, Pi-Prime.
Index entries for sequences related to "constant primes"
Index entries for sequences related to the number Pi

See A060421 for further terms.

Cf. A198018, A198019, A195834, A047777, A053013, A064467.

Digits := 130; n0 := evalf(Pi); for i from 1 to 120 do t1 := trunc(10^i*n0); if isprime(t1) then print(t1); fi; od:

a = {}; Do[k = Floor[Pi 10^n]; If[PrimeQ[k], AppendTo[a, k]], {n, 0, 160}]; a (* Artur Jasinski, Mar 26 2008 *)
nn=1000;With[{pidigs=RealDigits[Pi,10,nn][[1]]},Select[Table[FromDigits[ Take[pidigs,n]],{n,nn}],PrimeQ]] (* Harvey P. Dale, Sep 26 2012 *)

c=Pi;for(k=0,precision(c),isprime(c\.1^k) & print1(c\.1^k,",")) \\ - M. F. Hasler, Sep 01 2013

a(n) = floor(10^(A060421(n)-1)*A000796), where A000796 is the constant Pi = 3.14159... . - M. F. Hasler, Sep 02 2013

A019673 Decimal expansion of Pi/6.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

From Omar E. Pol, Aug 30 2007: (Start)
Pi/6 = Volume of the inscribed ellipsoid / (Volume of the cuboid (If L1>L2>L3)).
Pi/6 = Volume of the inscribed spheroid / (Volume of the cuboid (If L1>(L2=L3))).
Pi/6 = Volume of the inscribed spheroid / (Volume of the cuboid (If L1<(L2=L3))).
Pi/6 = Volume of the inscribed sphere / (Volume of the regular hexahedron (Or cube)).  (End)
Pi/6 = Surface area of the inscribed sphere / (surface area of the regular hexahedron (or cube)). - Omar E. Pol, Nov 13 2007
Decimal expansion of arctan(sqrt(1/3)). - Clark Kimberling, Sep 23 2011
Also, decimal expansion of sum( k>=1, (-120+329*k+568*k^2)/(k*(1+k)*(1+2*k)*(1+4*k)*(3+4*k)*(5+4*k)) ). - Bruno Berselli, Dec 01 2013
Atomic packing factor (APF) of the simple cubic lattice filled with spheres of the same diameter (unique example among chemical elements: polonium crystal). - Stanislav Sykora, Sep 29 2014

			Pi/6 = 0.5235987755982988730771072305465838140328615665625176368291574...

Ian Stewart, Professor Stewart's Cabinet of Mathematical Curiosities, Basic Books, a member of the Perseus Books Group, NY, 2009, "A Constant Bore", pp. 49-50 & 264-266.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Atomic packing factor
Index entries for transcendental numbers

Cf. A000796, A132696.

Cf. APF's of other crystal lattices: A093825 (hcp,fcc), A247446 (diamond cubic).

C := ComplexField(); [Pi(C)/6]; // G. C. Greubel, Nov 18 2017

Mathematica

RealDigits[N[Pi/6,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *) RealDigits[Pi/6,10,120][[1]] (* Harvey P. Dale, Oct 05 2024 *)

PARI

Pi/6 \\ Charles R Greathouse IV, Jul 07 2014

Formula

From Amiram Eldar, Aug 15 2020: (Start)

Equals Integral_{x=0..oo} 1/(x^2 + 9) dx.

Equals Integral_{x=0..oo} 1/(9*x^2 + 1) dx. (End)

Pi/6 = Sum_{n >= 1} i/(n*P(n,sqrt(-3))*P(n-1,sqrt(-3))), where i = sqrt(-1) and P(n,x) denotes the n-th Legendre polynomial. The first ten terms of the series gives the approximation Pi/6 = 0.52359877559(52...) correct to 11 decimal places - Peter Bala, Mar 16 2024

A037008 Positions of the digit '0' in the decimal expansion of Pi, where positions 0, 1, 2, ... correspond to digits 3, 1, 4, ....

Original entry on oeis.org

32, 50, 54, 65, 71, 77, 85, 97, 106, 116, 121, 128, 132, 146, 159, 164, 167, 176, 195, 207, 245, 248, 264, 270, 287, 291, 307, 308, 311, 327, 330, 340, 357, 360, 361, 366, 369, 375, 398, 403, 408, 421, 443, 451, 493, 513, 520, 523, 543, 545, 552, 557, 561

Offset: 1

Views

Graph

List

Refs

Listen

History

Text

Internal format

Author

Nicolau C. Saldanha (nicolau(AT)mat.puc-rio.br)

Keywords

base
nonn

Examples

Pi = 3.14159 26535 89793 23846 26433 83279 5*0*288 4... (Position 32 refers to the 32nd digit after the decimal point.)

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..369 from M. F. Hasler)

Eric Weisstein's World of Mathematics, Pi Digits.

Crossrefs

Cf. A000796 (decimal expansion (or digits) of Pi).

For another version see A014976(n) = a(n) + 1.

For digits 0 through 9 see: this sequence, A037000, A037001, A037002, A037003, A037004, A037005, A036974, A037006, A037007.

Programs

Mathematica

Flatten @ Position[ RealDigits[Pi - 3, 10, 500][[1]], 0] (* Robert G. Wilson v, Mar 07 2011 *)

PARI

for(c=1,default(realprecision,2011)-2,Pi\.1^c%10 || print1(c",")) \\ M. F. Hasler, Oct 23 2011

PARI

A037008_upto(N=999)={localprec(N+20); [i-1|i<-[1..#N=digits(Pi\10^-N)],!N[i]]} \\ M. F. Hasler, Jul 29 2024

Formula

a(n) = A014976(n) - 1. - M. F. Hasler, Jul 29 2024

Extensions

Name edited by M. F. Hasler, Jul 29 2024

A061360 Decimal expansion of e/Pi.

Original entry on oeis.org

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

Offset: 0

Views

Graph

List

Refs

Listen

History

Text

Internal format

Author

Jason Earls, Jun 07 2001

Keywords

cons
nonn

Examples

0.86525597943226508721777478964608961742874...

Links

Harry J. Smith, Table of n, a(n) for n = 0..20000

Crossrefs

Cf. A000796, A001113, A007526, A061382.

Programs

Mathematica

RealDigits[N[E/Pi,200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *) RealDigits[E/Pi,10,120][[1]] (* Harvey P. Dale, Sep 29 2023 *)

PARI

{ default(realprecision, 20080); x=10*exp(1)/Pi; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b061360.txt", n, " ", d)); } \\ Harry J. Smith, Jul 21 2009

Formula

Equals -1 + Product_{n>=0} (1 + 1/(A007526(n) + n!*Pi)). - David Ulgenes, Sep 21 2023

Extensions

Edited by N. J. A. Sloane, Sep 18 2008 at the suggestion of R. J. Mathar

A092425 Decimal expansion of Pi^4.

Original entry on oeis.org

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

Offset: 2

Views

Graph

List

Refs

Listen

History

Text

Internal format

Author

Mohammad K. Azarian, Mar 22 2004

Keywords

cons
easy
nonn

Examples

97.40909103400243723644033268870511124972758567268542169146785938997085...

Links

Harry J. Smith, Table of n, a(n) for n = 2..20000

Mohammad Reza Yegan, On the irrationality of Pi^4 and Pi^6, Journal of Number Theory, Volume 178, September 2017, Pages 5-10.

Index entries for transcendental numbers

Crossrefs

Cf. A000796 (Pi), A002388 (Pi^2), A091925 (Pi^3), A092731 (Pi^5), A092732 (Pi^6), A092735 (Pi^7), A092736 (Pi^8), A058286 (continued fraction), A013662.

Programs

Magma

R:= RealField(150); (Pi(R))^4; // G. C. Greubel, Mar 09 2018

Magma

R:=RealField(110); SetDefaultRealField(R); n:=Pi(R)^4; Reverse(Intseq(Floor(10^98*n))); // Bruno Berselli, Mar 12 2018

Mathematica

RealDigits[Pi^4, 10, 100][[1]] (* G. C. Greubel, Mar 09 2018 *)

PARI

default(realprecision, 20080); x=Pi^4/10; for (n=2, 20000, d=floor(x); x=(x-d)*10; write("b092425.txt", n, " ", d)); \\ Harry J. Smith, Jun 22 2009

Formula

Equals 120 * Sum_{j>=1} Sum_{i=1..j-1} 1/(i*j)^2. - Enrique Pérez Herrero, Jun 29 2012

Equals Sum_{k>=1} k*(k+1)*(k+2)*zeta(k+3)/2^(k-1). - Amiram Eldar, May 21 2021

From Peter Bala, Oct 21 2023: (Start)

Pi^4 = 90*Sum_{n >= 1} 1/n^4 (Euler).

The following faster converging series representations for the constant Pi^4 may be easily verified using partial fraction expansions of the summands of the series. Presumably, these are the first three cases of an infinite family of similar results.

Let P(n) = n*(n + 1)*(n + 2)/2!. Then Pi^4 = 1575/16 - 15*Sum_{n >= 1} d/dn(P(n))/P(n)^4.

Let Q(n) = n*(n + 1)*(n + 2)*(n + 3)*(n + 4)/4!. Then Pi^4 = 673165/6912 + Sum_{n >= 1} d/dn(Q(n))/Q(n)^4.

Let R(n) = n*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*(n + 6)/6!. Then Pi^4 = 5610787/57600 - (3/56)*Sum_{n >= 1} d/dn(R(n))/R(n)^4.

Taking 10 terms of the last series gives the approximation Pi^4 = 97.4090910340

024372(50...), correct to 16 decimal places. (End)

A229099 Decimal expansion of 1 - 6/Pi^2.

Original entry on oeis.org

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

Offset: 0

Views

Graph

List

Refs

Listen

History

Text

Internal format

Author

Charles R Greathouse IV, Sep 13 2013

Keywords

nonn
cons
easy

Comments

Probability that a random number is not squarefree; probability that two random numbers have a common divisor greater than 1.

Examples

0.39207289814597337133672322074163416657384735196652070692634580863496...

Links

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Index entries for transcendental numbers

Crossrefs

Cf. A000796, A002388, A013661, A059956, A063035.

Programs

Mathematica

RealDigits[1-1/Zeta[2],10,90][[1]] (* Stefano Spezia, Feb 21 2025 *)

PARI

1-6/Pi^2

Formula

Equals 1 - 1/zeta(2). - Stefano Spezia, Feb 21 2025

Previous Showing 61-70 of 1164 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.
Content is available under The OEIS End-User License Agreement.

cp's OEIS Frontend

A197682 Decimal expansion of Pi/(2 + 2*Pi).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A004602 Expansion of Pi in base 3.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A063438 a(n) = floor((n+1)*Pi) - floor(n*Pi).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

Extensions

A086201 Decimal expansion of 1/(2*Pi).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A005042 Primes formed by the initial digits of the decimal expansion of Pi.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A019673 Decimal expansion of Pi/6.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A037008 Positions of the digit '0' in the decimal expansion of Pi, where positions 0, 1, 2, ... correspond to digits 3, 1, 4, ....

Views

A063438 a(n) = floor((n+1)Pi) - floor(nPi).