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 A214533 #20 Aug 21 2023 11:38:22
%S A214533 2,5,4,9,1,2,7,7,2,9,3,7,9,1,6,7,4,0,7,5,8,1,9,6,7,0,2,8,4,9,6,4,2,5,
%T A214533 4,0,5,4,7,1,4,5,0,8,4,6,3,8,4,8,6,4,0,4,1,7,4,5,5,6,3,5,5,1,1,3,2,1,
%U A214533 6,3,7,1,4,8,6,0,9,8,8,6,5,1,1,5,3,1
%N A214533 Decimal expansion of 1/2 + 2/sqrt(3) + 2/sqrt(5).
%C A214533 The convergent of a sum of reciprocals of square roots with numerators equal to the numerators in the Dirichlet series for Mangoldt Lambda [6] = 0.
%C A214533 Superposition of Dirichlet series of 6 shifted versions of A100051 evaluated at s=1/2.
%C A214533 The nontrivial Riemann zeta zeros are known to not be multiples of any number. This number -2.5491277293... comes close to relating the 18th, 33rd and 42nd zeta zeros to the first, second, and third zeta zeros, respectively.
%C A214533 ZetaZero[18]/2/2.549127729379167407581
%C A214533 ZetaZero[1]
%C A214533 14.135650568603255663
%C A214533 14.134725141734693790
%C A214533 ZetaZero[33]/2/2.549127729379167407581
%C A214533 ZetaZero[2]
%C A214533 21.020643640006420723
%C A214533 21.022039638771554993
%C A214533 ZetaZero[42]/2/2.549127729379167407581
%C A214533 ZetaZero[3]
%C A214533 25.011827067342131577
%C A214533 25.010857580145688763
%C A214533 Numerators in the sum for this constant are the sixth row and column in matrix A191898. The increment in the denominators is equal to 1, and the denominators begin:
%C A214533 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 4, 5, 6, 7, 8, 9, 5, 6, 7, 8, 9, 10, 6, 7, 8, 9, 10, 11, ...
%C A214533 Sums of this type that have numerators equal to Dirichlet series for logarithms are partials sums of square roots.
%C A214533 An algebraic number of degree 4 and denominator 30; minimal polynomial 3600x^4 - 7200x^3 - 9960x^2 + 13560x - 2591. - _Charles R Greathouse IV_, Apr 21 2016
%H A214533 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%F A214533 Equals the absolute value of sum_{n=1..infinity} [1/(n + 0)^(1/2) - 1/(n + 1)^(1/2) - 2/(n + 2)^(1/2) - 1/(n + 3)^(1/2) + 1/(n + 4)^(1/2) + 2/(n + 5)^(1/2)]
%t A214533 RealDigits[1/2+2/Sqrt[3]+2/Sqrt[5],10,120][[1]] (* _Harvey P. Dale_, Jul 31 2013 *)
%o A214533 (PARI) 1/2 + 2/sqrt(3) + 2/sqrt(5) \\ _Charles R Greathouse IV_, Mar 10 2016
%Y A214533 Cf. A191898.
%K A214533 nonn,cons
%O A214533 1,1
%A A214533 _Mats Granvik_, Jul 20 2012
%E A214533 Corrected by _Harvey P. Dale_, Jul 31 2013