A386718 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.

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

Offset

0,1

References

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See section 2.43, page 106.

Links

Table of n, a(n) for n=0..104.

Yaming Yu, A Multiple Integral in Terms of Stieltjes Constants, SIAM Problems and Solutions, Classical Analysis, Integrals, Problem 07-002 (2007).

Formula

Equals 1 - gamma - gamma_1 - gamma_2/2, where gamma_k is the k-th Stieltjes constant.

In general, for m >= 1, Integral_{x_1=0..1} ... Integral_{x_m=0..1} {1/(x_1*...*x_m)} dx_1 ... dx_m = 1 - Sum_{k=0..m-1} gamma_k/k!, where gamma_0 = gamma is Euler's constant.

Example

0.50044536217858002349633947881010515277510990544508...

Mathematica

With[{m = 2}, RealDigits[1 - Sum[StieltjesGamma[k]/k!, {k, 0, 2}], 10, 120][[1]]]

Crossrefs

Cf. A001620 (gamma), A082633 (-gamma_1), A086279 (-gamma_2).

Cf. A153810 (m=1), A242610 (m=2), this constant (m=3).

Sequence in context: A193524 A300237 A105077 * A073231 A099223 A233427

Adjacent sequences: A386715 A386716 A386717 * A386719 A386720 A386721

Keyword

nonn,cons

Author

Amiram Eldar, Jul 31 2025

Status

approved