A386733 -id:A386733 - cp's OEIS frontend

A386734 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} {1/(x+y+z)} dx dy dz, where {} denotes fractional part.

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

Offset: 0

Amiram Eldar, Aug 01 2025

			0.18384376406702461207534175664658126707821355787059...

Ovidiu Furdui, Multiple Fractional Part Integrals and Euler's Constant, Miskolc Mathematical Notes, Vol. 17, No. 1 (2016), pp. 255-266.

Cf. A002117, A002162, A002391, A153810, A386733, A386736, A386737.

RealDigits[9*Log[3]/2 - 6*Log[2] - Zeta[3]/2, 10, 120][[1]]

PARI
```
9*log(3)/2 - 6*log(2) - zeta(3)/2
```

Equals 9*log(3)/2 - 6*log(2) - zeta(3)/2.

A386735 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} {1/(x+y)}^2 dx dy, where {} denotes fractional part.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Aug 01 2025

			0.40717012111440861174004820513640840627286557909642...

Ovidiu Furdui, Exotic fractional part integrals and Euler's constant, Analysis, Vol. 31 (2011), pp. 249-257.
Huizeng Qin and Youmin Lu, Integrals of Fractional Parts and Some New Identities on Bernoulli Numbers, Int. J. Contemp. Math. Sciences, Vol. 6, No. 15 (2011), pp. 745-761. See eq. (3.1). Note that this equation has an error, 3/2 instead of 5/2.

Cf. A001620 (gamma), A002162, A072691, A386733 (m=1).

RealDigits[5/2 - Log[2] - EulerGamma - Pi^2/12, 10, 120][[1]]

PARI
```
5/2 - log(2) - Euler - Pi^2/12
```

Equals 5/2 - log(2) - gamma - Pi^2/12.

For m >= 3, Integral_{x=0..1} Integral_{y=0..1} {1/(x+y)}^m dx dy = (2^(2-m) + m - 3)/((m-1)*(m-2)) + (m!/2) * Sum_{j>=1} ((j+1)!/(m+j)!) * (zeta(j+2) - 1).

cp's OEIS Frontend

A386734 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} {1/(x+y+z)} dx dy dz, where {} denotes fractional part.

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