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 A386716 #10 Jul 31 2025 10:55:56
%S A386716 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,
%T A386716 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,
%U A386716 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
%N A386716 Decimal expansion of Integral_{x=0..1} {1/x}^3 * x^3 dx, where {} denotes fractional part.
%D A386716 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 A386716 Equals 1 - (zeta(2) + zeta(3) + zeta(4))/4.
%F A386716 Equals Integral_{x=0..1} Integral_{y=0..1} {x/y}^3 * {y/x}^3 dx dy.
%F A386716 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 A386716 0.01767144907026027165296074382533922931032821363349...
%t A386716 RealDigits[1 - (Zeta[2] + Zeta[3] + Zeta[4])/4, 10, 120, -1][[1]]
%o A386716 (PARI) 1 - (zeta(2) + zeta(3) + zeta(4))/4
%Y A386716 Cf. A002117, A013661, A013662.
%Y A386716 Cf. A354238 (m=1), A386715 (m=2), this constant (m=3).
%K A386716 nonn,cons
%O A386716 0,3
%A A386716 _Amiram Eldar_, Jul 31 2025