A345208 -id:A345208 - cp's OEIS frontend

A386733 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} {1/(x+y)} dx dy, where {} denotes fractional part.

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

Offset: 0

Amiram Eldar, Aug 01 2025

			0.56382732769577740059825665959334054154152531811711...

Ovidui Furdui, Problem 150, Problems, Missouri J. Math. Sci., Vol. 16, No. 2 (2004), p. 130; Huizeng Qin, Solution to Problem 150, ibid., Vol. 17, No. 3 (2005), pp. 197-199.
Ovidiu Furdui, Multiple Fractional Part Integrals and Euler's Constant, Miskolc Mathematical Notes, Vol. 17, No. 1 (2016), pp. 255-266.

Cf. A016627, A072691, A153810, A345208, A386734, A386735.

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

PARI
```
2*log(2) - zeta(2)/2
```

Equals 2*log(2) - Pi^2/12 = A016627 - A072691.

A386713 Decimal expansion of Integral_{x=0..1} {1/x}^2 * {1/(1-x)}^2 dx, where {} denotes fractional part.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jul 31 2025

			0.04264560603125049181658953091533139472254244534257...

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See section 2.11, page 101.

Ovidiu Furdui, A class of fractional part integrals and zeta function values, Integral Transforms and Special Functions, Vol. 24, No. 6 (2013), pp. 485-490.

Cf. A001620 (gamma), A061444, A345208.

Cf. A147533 (m=1), this constant (m=2), A386714 (m=3).

RealDigits[4*Log[2*Pi] - 4*EulerGamma - 5, 10, 120, -1][[1]]

PARI
```
4*log(2*Pi) - 4*Euler - 5
```

Equals 4*log(2*pi) - 4*gamma - 5.
Equals 4*A345208 - 1.
In general, for m >= 2, Integral_{x=0..1} {1/x}^m * {1/(1-x)}^m dx = 2 * (Sum_{j=2..m-1} (-1)^(m+j-1) * (zeta(j)-1)) + (-1)^m - (2*m) * Sum_{k>=0} (zeta(2*k+m) - zeta(2*k+m+1))/(k+m) (note that the first sum vanishes when m = 2).

A369633 Decimal expansion of integral of frac(1/x)^3 dx for x=0 to 1.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Jan 28 2024

			0.18707307250977978945095915767776663195781480296221...

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals, Problems in Mathematical Analysis, Springer, 2013. See p. 100.

Cf. A074962, A345208.

RealDigits[3*Log[2*Pi]/2 - 6*Log[Glaisher] - EulerGamma - 1/2, 10, 120][[1]] (* Amiram Eldar, Jan 28 2024 *)

3*log(2*Pi)/2 + 6*zeta'(-1) - Euler - 1 \\ Amiram Eldar, Jan 28 2024

Integral_{x=0..1} frac(1/x)^3 dx = (3/2)*log(2*Pi) - 6*log(A) - gamma - 1/2 = 0.1870730725..., where A is the Glaisher-Kinkelin constant.
Equals 3*log(2) - 3/2 + 3 * Sum_{k>=1} ((-1)^k/(k+3))*(zeta(k+1)-1).
From Vaclav Kotesovec, Jan 29 2024: (Start)
Equals 6 * Sum_{k>=1} (zeta(k+1) - 1) / ((k+1)*(k+2)*(k+3)).
Equals -1/2 + 6 * Sum_{k>=2} zeta(k) / (k*(k+1)*(k+2)). (End)

A386738 Decimal expansion of Integral_{x=0..1} {1/x}^4 dx, where {} denotes fractional part.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Aug 01 2025

			0.14553289487913287197745596494722440166566646379514...

Ovidiu Furdui, Problem 3366, Crux Mathematicorum, Vol. 34, No. 6 (2008), pp. 362 and 365; Solution to Problem 3366, by Chip Curtis, ibid., Vol. 35, No. 6 (2009), pp. 403-405.
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.

Cf. A001620 (gamma), A002117, A061444, A073002, A306016.

Cf. A153810 (m=1), A345208 (m=2), A345208 (m=3), this constant (m=4).

RealDigits[Log[2*Pi] - 2*EulerGamma - 1/3 + (Zeta[3]/2 + Zeta'[2])/Zeta[2], 10, 120][[1]]

log(2*Pi) - 2*Euler - 1/3 + (zeta(3)/2 + zeta'(2))/zeta(2)

Equals log(2*Pi) - 2*gamma - 1/3 + 3*zeta(3)/Pi^2 + 6*zeta'(2)/Pi^2.

In general, for m >= 2, Integral_{x=0..1} {1/x}^m dx = log(2*Pi) - m*gamma/2 - 1/(m-1) - Sum_{k=1..floor((m-2)/2)} (-1)^k * (m!/(m-2*k-1)!) * zeta(2*k+1) / (2^(2*k+1) * Pi^(2*k)) + 2 * Sum_{k=1..floor((m-1)/2)} (-1)^(k-1) * (m!/(m-2*k)!) * zeta'(2*k) / (2*Pi)^(2*k).

cp's OEIS Frontend

A386733 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} {1/(x+y)} dx dy, where {} denotes fractional part.

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A386713 Decimal expansion of Integral_{x=0..1} {1/x}^2 * {1/(1-x)}^2 dx, where {} denotes fractional part.

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A369633 Decimal expansion of integral of frac(1/x)^3 dx for x=0 to 1.

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A386738 Decimal expansion of Integral_{x=0..1} {1/x}^4 dx, where {} denotes fractional part.

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