A386715 Decimal expansion of Integral_{x=0..1} {1/x}^2 * x^2 dx, where {} denotes fractional part.

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

Offset

0,2

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))/3.

Equals 1 - A347213 / 3.

Equals Integral_{x=0..1} Integral_{y=0..1} {x/y}^2 * {y/x}^2 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

0.05100300999739309270928222394750827333868793548423...

Mathematica

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

Prog

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

Crossrefs

Cf. A002117, A013661, A347213.

Cf. A354238 (m=1), this constant (m=2), A386716 (m=3).

Sequence in context: A269129 A320606 A343016 * A058177 A345373 A204619

Adjacent sequences: A386712 A386713 A386714 * A386716 A386717 A386718

Keyword

nonn,cons

Author

Amiram Eldar, Jul 31 2025

Status

approved