cp's OEIS Frontend

A358045 Decimal expansion of 2*(gamma + Re(Psi(i))).

%I A358045 #52 Jun 25 2023 08:09:56
%S A358045 1,3,4,3,7,3,1,9,7,1,0,4,8,0,1,9,6,7,5,7,5,6,7,8,1,1,4,5,6,0,8,6,2,6,
%T A358045 3,0,7,0,3,6,8,4,4,6,1,5,4,0,6,9,3,0,4,4,4,0,7,7,5,1,3,9,1,8,0,0,7,5,
%U A358045 4,5,6,8,3,0,7,3,8,9,0,6,4,8,6,4,0,8,3
%N A358045 Decimal expansion of 2*(gamma + Re(Psi(i))).
%C A358045 Here gamma = 0.5772... (A001620) is Euler's constant and Re(Psi(i)) = 0.0946... (A248177) with the psi function or the digamma function Psi(x) = d/dx log(Gamma(x)) = (1/Gamma(x))*d/dx Gamma(x) as the logarithmic derivative of the gamma function.
%C A358045 Value of the series 1/1 + 1/(2+3) + 1/(4+5+6) + 1/(7+8+9+10) + ... = Sum_{k>=1} 2/(k*(k^2+1)) = Sum_{k>=1} 1/A006003(k), i.e., the sum of the reciprocals of the sum of the next n natural numbers. Presumably, the question of finding the value of this series was first posed in 1889 by the Dutch mathematician Pieter Hendrik Schoute (1846-1913) and repeated in 1905.
%H A358045 Prof. Schoute, <a href="https://books.google.de/books?id=n5BAAQAAMAAJ&amp;pg=PA293#v=onepage&amp;q&amp;f=false">Question 10184</a>, The Educational Times, and Journal of the College of Preceptors 42 (1889), nr. 339 (July 1), p. 293; <a href="https://archive.org/details/educationaltimes58educ/page/534/mode/1up?view=theater">Question 10184</a>, Ibid., 58 (1905), nr. 536 (Dec. 1), p. 534.
%e A358045 1.343731971048019675756781145608...
%p A358045 2*(gamma+Re(Psi(I))); evalf(%,200);
%t A358045 RealDigits[2*(EulerGamma + Re[PolyGamma[0, I]]), 10, 120][[1]] (* _Amiram Eldar_, Dec 20 2022 *)
%Y A358045 Cf. A001620, A006003, A248177.
%Y A358045 Essentially the same as A340012.
%K A358045 nonn,cons
%O A358045 1,2
%A A358045 _Martin Renner_, Dec 20 2022