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A386738
Decimal expansion of Integral_{x=0..1} {1/x}^4 dx, where {} denotes fractional part.
0
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
0,2
LINKS
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.
FORMULA
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).
EXAMPLE
0.14553289487913287197745596494722440166566646379514...
MATHEMATICA
RealDigits[Log[2*Pi] - 2*EulerGamma - 1/3 + (Zeta[3]/2 + Zeta'[2])/Zeta[2], 10, 120][[1]]
PROG
(PARI) log(2*Pi) - 2*Euler - 1/3 + (zeta(3)/2 + zeta'(2))/zeta(2)
CROSSREFS
Cf. A153810 (m=1), A345208 (m=2), A345208 (m=3), this constant (m=4).
Sequence in context: A120651 A093054 A029685 * A153131 A016718 A165361
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Aug 01 2025
STATUS
approved