A074950 -id:A074950 - cp's OEIS frontend

A197733 Decimal expansion of 2*Pi/(1+Pi).

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

Offset: 1

Clark Kimberling, Oct 17 2011

Least x>0 such that sin(bx)=cos(cx) (and also sin(cx)=cos(bx)), where b=1/4 and c=Pi/4; see the Mathematica program for a graph and A197682 for a discussion and guide to related sequences.

Equals the harmonic mean of 1 and Pi. - Stanislav Sykora, Apr 11 2016

			1.51709398598955229068886137808978572827685273...

Index entries for transcendental numbers

Cf. A074950 (harmonic mean of Pi and e), A197682.

MATLAB
```
2*pi/(1+pi) % Altug Alkan, Apr 11 2016
```

b = 1/4; c = Pi/4;
t = x /. FindRoot[Sin[b*x] == Cos[c*x], {x, 1.517, 1.518}]
N[Pi/(2*b + 2*c), 110]
RealDigits[%]  (* A197733 *)
Simplify[Pi/(2*b + 2*c)]
Plot[{Sin[b*x], Cos[c*x]}, {x, 0, Pi/2}]
RealDigits[2Pi/(1+Pi),10,120][[1]] (* Harvey P. Dale, Jun 17 2022 *)

2*Pi/(1+Pi) \\ Michel Marcus, Apr 11 2016

A074951 Continued fraction for harmonic mean of Pi and e.

Original entry on oeis.org

2, 1, 10, 1, 2, 1, 1, 10, 1, 1, 2, 1, 11, 1, 5, 1, 1, 7, 58, 3, 1, 2, 2, 7, 13, 3, 1, 20, 1, 20, 1, 2, 8, 2, 1, 1, 2, 1, 1, 14, 3, 4, 1, 1, 1, 1, 8, 1, 6, 4, 1, 3, 1, 2, 2, 1, 1, 12, 42, 9, 1, 1, 1, 1, 6, 1, 1, 3, 8, 2, 7, 1, 1, 2, 2, 1, 1, 3, 1, 17, 1, 3, 2, 4, 1, 2, 1, 4, 3, 1, 1, 1, 7, 13, 22, 1, 2

Offset: 0

Zak Seidov, Oct 05 2002

Cf. A074950 (decimal expansion).

ContinuedFraction[HarmonicMean[{Pi,E}],100] (* Harvey P. Dale, May 16 2013 *)

Offset changed by Andrew Howroyd, Jul 06 2024

A342209 Decimal expansion of logarithmic mean of Pi and e.

Original entry on oeis.org

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

Offset: 1

Gleb Koloskov, Mar 05 2021

The logarithmic mean of Pi and e lies between the arithmetic and geometric means of Pi and e.

			2.9248335452376948992733592...

B. C. Carlson, The logarithmic mean, The American Mathematical Monthly, 79(6) 1972, pp. 615-618.
Wikipedia, Logarithmic mean.

Cf. A000796 (Pi), A001113 (e), A019645, A074916, A074948, A074950, A074957.

RealDigits[(Pi - E)/(Log[Pi] - 1), 10, 100][[1]] (* Amiram Eldar, Mar 06 2021 *)

PARI
```
(Pi-exp(1))/(log(Pi)-1)
```

Equals (Pi-e)/(log(Pi)-1).

cp's OEIS Frontend

A197733 Decimal expansion of 2*Pi/(1+Pi).

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A074951 Continued fraction for harmonic mean of Pi and e.

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A342209 Decimal expansion of logarithmic mean of Pi and e.

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