A386716 Decimal expansion of Integral_{x=0..1} {1/x}^3 * x^3 dx, where {} denotes fractional part.

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

Offset

0,3

References

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See section 2.21, pages 103 and 110.

Links

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

Formula

Equals 1 - (zeta(2) + zeta(3) + zeta(4))/4.

Equals Integral_{x=0..1} Integral_{y=0..1} {x/y}^3 * {y/x}^3 dx dy.

In general, for m >= 1, Integral_{x=0..1} {1/x}^m * x^m dx = Integral_{x=0..1} Integral_{y=0..1} {x/y}^m * {y/x}^m dx dy = 1 - Sum_{k=2..m+1} zeta(k)/(m+1).

Example

0.01767144907026027165296074382533922931032821363349...

Mathematica

RealDigits[1 - (Zeta[2] + Zeta[3] + Zeta[4])/4, 10, 120, -1][[1]]

Prog

(PARI) 1 - (zeta(2) + zeta(3) + zeta(4))/4

Crossrefs

Cf. A002117, A013661, A013662.

Cf. A354238 (m=1), A386715 (m=2), this constant (m=3).

Sequence in context: A388327 A242977 A247314 * A257395 A197687 A239341

Adjacent sequences: A386713 A386714 A386715 * A386717 A386718 A386719

Keyword

nonn,cons

Author

Amiram Eldar, Jul 31 2025

Status

approved