A092425 -id:A092425 - cp's OEIS frontend

A060302 Decimal expansion of (Pi^4 + Pi^5)^(1/6).

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

Offset: 1

Jason Earls, Mar 26 2001

Approximation to e, accurate to 7 decimal places.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for transcendental numbers.

Cf. A001113, A000796, A092425, A092731.

RealDigits[Surd[Pi^4 + Pi^5, 6], 10, 120][[1]] (* Amiram Eldar, Jun 13 2023 *)

default(realprecision, 20080); x=(Pi^4 + Pi^5)^(1/6); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b060302.txt", n, " ", d)); \\ Harry J. Smith, Jul 03 2009

A196751 Decimal expansion of 8*Pi^4/729.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Oct 06 2011

			1.0689612184801364854479048854727584224935811870802241063535...

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
L. B. W. Jolley, Summation of Series, Dover, 1961. Eq 309 at n=4.
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015.
Index entries for transcendental numbers

Cf. A092425.

RealDigits[8(Pi^4/729), 10, 90][[1]] (* Bruno Berselli, Dec 20 2011 *)

8*Pi^4/729 \\ Charles R Greathouse IV, Oct 06 2011

Equals Sum_{n>=1} A011655(n)/n^4. See Mathar link, L(m=3,r=1,s=4).

A257134 Decimal expansion of Pi^4/45.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 16 2015

			2.16464646742227638303200739308233580554950190383745381536595243...

L. J. P. Kilford, Modular Forms: A Classical and Computational Introduction, Imperial College Press, 2008, p. 15.

Alain Tissier, Apéry's Constant, Solution to Problem 10635, The American Mathematical Monthly, Vol. 106, No. 10 (1999), pp. 965-966.
Eric Weisstein's World of Mathematics, Eisenstein Series.
Index entries for transcendental numbers.

Cf. A013662, A092425, A231535, A257136.

Mathematica
```
RealDigits[Pi^4/45, 10, 105] // First
```

Pi^4/45 \\ Charles R Greathouse IV, Oct 01 2022

Pi^4/45 = 2*zeta(4) = G_4(oo), where the function G_k(z) is the Eisenstein nonzero modular form of weight k.
Equals -Integral_{x=0..1} log(x)^2 * log(1 - x)/x dx. - Amiram Eldar, Jul 21 2020
Equals Sum_{n,m>=1} (Pi^2/6 - Sum_{k=1..n+m} 1/k^2)/(n*m) (Tissier, 1999). - Amiram Eldar, Jan 27 2024
Equals Integral_{x=0..1} Li(3,sqrt(x))/x dx, where Li(n,x) is the polylogarithm function. - Kritsada Moomuang, Jun 18 2025
Equals 2*A013662 = A231535/3. - Hugo Pfoertner, Jun 18 2025

A383647 Decimal expansion of 15/(2*Pi^4).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, May 03 2025

			0.07699486691013251391864587450339020606370851390...

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 64.

Jakub Buczak, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers.

Cf. A030059, A082020, A088245, A088246, A092425, A151927.

Join[{0},RealDigits[15/(2Pi^4),10,100][[1]]]

Equals Sum_{n > 0} 1/A030059(n)^4.

Equals 10/A151927. - Hugo Pfoertner, May 03 2025

A194657 Decimal expansion of (4Pi^6log(2) - 90Pi^4zeta(3) + 1350Pi^2zeta(5) - 5715*zeta(7))/1536.

Original entry on oeis.org

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

Offset: 0

Seiichi Kirikami, Sep 01 2011

The absolute value of the integral {x=0..Pi/2} x^5*log(sin(x )) dx or (d^5/da^5 (integral {x=0..Pi/2} sin(ax)*log(sin(x )) dx)) at a=0. The absolute value of m=2 of (-1)^(m+1)*(sum {n=1..infinity} (limit {a -> 0} (d^(2m+1)/da^(2m+1) ((1-cos((a+2n)*Pi/2))/n/(a+2n)))))-(pi/2)^2(m+1)*log(2)/2/(m+1).

			0.11757583407233248206...

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 1.441.2

Cf. A173623, A173624, A193716, A193717, A196456.

RealDigits[ N[(4 Pi^6*Log[2]-90 Pi^4*Zeta[3]+1350 Pi^2*Zeta[5]-5715 Pi^2*Zeta[7])/1536,150]][[1]]

Equals (4*A092732*A002162-90*A092425*A002117+1350*A002388*A013663-5715*A013665)/1536.

A277117 Decimal expansion of e^6/(Pi^5+Pi^4), where e = exp(1).

Original entry on oeis.org

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

Offset: 1

Paulo Romero Zanconato Pinto, Sep 30 2016

			1.0000000438081076299476374320122890058190409215306036959233520...

D. Wilson, pers. comm.

Eric Weisstein's World of Mathematics, Almost Integer
Wikipedia, Near-identities involving e and pi.

Cf. A000796, A001113, A092425, A092731, A092512, A239606.

RealDigits[E^6/(Pi^5+Pi^4),10,120][[1]] (* Harvey P. Dale, Apr 07 2019 *)

exp(6)/(Pi^5+Pi^4) \\ Michel Marcus, Oct 02 2016

Equals A092512 /(A092731 + A092425).

A376994 Decimal expansion of Pi^4 + Pi^5.

Original entry on oeis.org

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

Offset: 3

Stefano Spezia, Oct 12 2024

This constant is very close to e^6 = A092512.

			403.4287758192838904991816427321407177300282923...

Index entries for transcendental numbers.

Cf. A092425, A092512, A092731.

Mathematica
```
RealDigits[Pi^4+Pi^5,10,100][[1]]
```

A377647 Decimal expansion of Pi^4/15 + Pi^2log(2)^2/4 - log(2)^4/4 - 21log(2)zeta(3)/4 - 6Li_4(1/2).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 03 2024

			0.142514197935710915708721501209652160895463410764...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.8, p. 47.

Michael I. Shamos, A catalog of the real numbers, (2007). See pp. 181-182.

Cf. A002117, A002162, A002388, A092425, A099218.

RealDigits[Pi^4/15+Pi^2Log[2]^2/4-Log[2]^4/4-21Log[2]Zeta[3]/4-6PolyLog[4,1/2],10,100][[1]]

Integral_{x=1..2} log(x)^3/(x - 1) [Levin] (see Finch).

Integral_{x=0..1} log(1 + x)^3/x (see Shamos).

cp's OEIS Frontend

A060302 Decimal expansion of (Pi^4 + Pi^5)^(1/6).

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A196751 Decimal expansion of 8*Pi^4/729.

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A257134 Decimal expansion of Pi^4/45.

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A383647 Decimal expansion of 15/(2*Pi^4).

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A194657 Decimal expansion of (4*Pi^6*log(2) - 90*Pi^4*zeta(3) + 1350*Pi^2*zeta(5) - 5715*zeta(7))/1536.

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A277117 Decimal expansion of e^6/(Pi^5+Pi^4), where e = exp(1).

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A376994 Decimal expansion of Pi^4 + Pi^5.

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A377647 Decimal expansion of Pi^4/15 + Pi^2*log(2)^2/4 - log(2)^4/4 - 21*log(2)*zeta(3)/4 - 6*Li_4(1/2).

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A194657 Decimal expansion of (4Pi^6log(2) - 90Pi^4zeta(3) + 1350Pi^2zeta(5) - 5715*zeta(7))/1536.

A377647 Decimal expansion of Pi^4/15 + Pi^2log(2)^2/4 - log(2)^4/4 - 21log(2)zeta(3)/4 - 6Li_4(1/2).