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 A147533 #29 Jan 05 2025 19:51:39
%S A147533 1,5,4,4,3,1,3,2,9,8,0,3,0,6,5,7,2,1,2,1,3,0,2,4,1,8,0,1,6,4,8,0,4,8,
%T A147533 6,2,0,8,4,3,1,8,6,7,1,8,7,9,8,4,7,1,9,7,6,1,1,5,3,4,4,6,9,7,6,9,7,3,
%U A147533 5,4,5,3,5,5,5,3,2,9,3,4,1,8,7,3,8,9,4,1,2,6,5,8,3,4,9,3,4,9,9,0,2,9,2,6
%N A147533 Decimal expansion of 2*gamma-1, where gamma is the Euler-Mascheroni constant.
%C A147533 This constant arises in the dynamical system: z(n+1) = F(z(n)) where F(z) = conjugate(zeta(z))/zeta(z) and zeta is the Riemann zeta function. For instance, starting from z(0) with Re(z(0)) > 0 and Im(z(0)) > 0 we get lim_{n->oo} Im(z(2n))/Im(z(2n-1)) = 2*gamma - 1. See Finch's book p. 29 for another appearance.
%D A147533 Steven R. Finch, "Euler-Mascheroni constant, gamma", Section 1.5 in Mathematical Constants, Cambridge University Press, 2003, pp. 28-32.
%D A147533 Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See Problem 2.10, pages 101 and 115-116.
%H A147533 G. C. Greubel, <a href="/A147533/b147533.txt">Table of n, a(n) for n = 0..10000</a>
%H A147533 Ovidiu Furdui, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/AdvProbSlnMay2016.pdf">Problem H-790</a>, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 54, No. 2 (2016), p. 186; <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/AdvProbMay2018.pdf">A series involving harmonic numbers and the zeta function at positive integers</a>, Solution to Problem H-790 by Ramya Dutta, ibid., Vol. 56, No. 2 (2018), pp. 190-191.
%F A147533 Equals limit_{n->oo} ((1/n)*Sum_{k=1..n} tau(k) - log(n)), where tau(k) is the number of divisors of k. - _Jean-François Alcover_, Mar 28 2013, after S. R. Finch's book, p. 29.
%F A147533 Equals Sum_{k>=2} (k-2)*(zeta(k)-1)/k. - _Amiram Eldar_, Jun 26 2021
%F A147533 Equals Sum_{k>=2} (H(k) - gamma - Sum_{j=2..k} zeta(j)/j), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Furdui, 2016). - _Amiram Eldar_, Dec 02 2021
%F A147533 Equals Integral_{x=0..1} frac(1/x)*frac(1/(1-x)) dx (Furdui, 2013). - _Amiram Eldar_, Mar 26 2022
%e A147533 2*gamma - 1 = 0.15443132980306572121302418016480486208431867187984...
%p A147533 evalf(2*gamma-1); # _R. J. Mathar_, Jan 26 2009
%t A147533 RealDigits[2*EulerGamma - 1, 10, 100][[1]] (* _G. C. Greubel_, Aug 31 2018 *)
%o A147533 (PARI) 2*Euler-1
%o A147533 (Magma) R:= RealField(100); 2*EulerGamma(R) -1; // _G. C. Greubel_, Aug 31 2018
%Y A147533 Cf. A000005, A001008, A001620, A002805.
%K A147533 cons,nonn
%O A147533 0,2
%A A147533 _Benoit Cloitre_, Nov 06 2008