This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A386715 #10 Jul 31 2025 10:55:07
%S A386715 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,
%T A386715 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,
%U A386715 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
%N A386715 Decimal expansion of Integral_{x=0..1} {1/x}^2 * x^2 dx, where {} denotes fractional part.
%D A386715 Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See section 2.21, pages 103 and 110.
%F A386715 Equals 1 - (zeta(2) + zeta(3))/3.
%F A386715 Equals 1 - A347213 / 3.
%F A386715 Equals Integral_{x=0..1} Integral_{y=0..1} {x/y}^2 * {y/x}^2 dx dy.
%F A386715 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).
%e A386715 0.05100300999739309270928222394750827333868793548423...
%t A386715 RealDigits[1 - (Zeta[2] + Zeta[3])/3, 10, 120, -1][[1]]
%o A386715 (PARI) 1 - (zeta(2) + zeta(3))/3
%Y A386715 Cf. A002117, A013661, A347213.
%Y A386715 Cf. A354238 (m=1), this constant (m=2), A386716 (m=3).
%K A386715 nonn,cons
%O A386715 0,2
%A A386715 _Amiram Eldar_, Jul 31 2025