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

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, 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, 4, 5, 6, 8, 3, 0, 7, 3, 8, 9, 0, 6, 4, 8, 6, 4, 0, 8, 3

Offset: 1

Author

Martin Renner, Dec 20 2022

Keywords

• nonn
• cons

Comments

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.

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.

Examples

			1.343731971048019675756781145608...
		

Links

• Prof. Schoute, Question 10184, The Educational Times, and Journal of the College of Preceptors 42 (1889), nr. 339 (July 1), p. 293; Question 10184, Ibid., 58 (1905), nr. 536 (Dec. 1), p. 534.

Crossrefs

Cf. A001620, A006003, A248177.

Essentially the same as A340012.

Programs

Maple

2*(gamma+Re(Psi(I))); evalf(%,200);

Mathematica

RealDigits[2*(EulerGamma + Re[PolyGamma[0, I]]), 10, 120][[1]] (* Amiram Eldar, Dec 20 2022 *)