A389850 - OEIS

%I #31 Nov 09 2025 16:44:45

%S 1,0,4,4,3,3,1,4,9,3,6,8,8,0,2,8,4,7,5,2,1,0,8,9,4,5,0,2,4,6,1,3,5,1,

%T 4,1,3,5,3,5,4,8,9,2,4,2,3,9,6,4,6,1,6,5,2,2,9,4,7,7,5,7,7,3,1,7,2,9,

%U 6,0,9,5,9,0,6,8,5,5,9,4,3,1,7,4,8,6,2,8

%N Decimal expansion of the absolute value of the sum of gamma function at all its extremas.

%C The extremes occur at each x_i for which digamma(x_i) = 0.

%C These Gamma(x_i) values alternate in sign and decrease in magnitude, so their sum converges.

%C Let t(n) = arctan(Pi/log(n))/Pi then the above constant approximately equals sqrt(Pi)/2 + Sum_{i>=1} Gamma(t(i)-i) = sqrt(Pi)/2 + Sum_{i>=1} ((-1)^i) * sqrt(log(i)^2+Pi^2) / (Gamma(1+i-t(i))) which agree to 4 digits after the decimal point.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Digamma_function#Roots_of_the_digamma_function">Digamma Roots</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Gamma_function#Minima_and_maxima">Gamma Extrema</a>.

%F Equals Sum_{n>=0} Gamma(x_n) where x_n is n-th zero of the digamma function.

%e abs(-1.04433149368802847521089...).

%e The sum begins 0.885603... + (-3.544643) + 2.302407 + ... (being A030171 at x_0=A030169, then Gamma(A175472), Gamma(A175473), ...).

%Y Cf. A030171, A175472, A175473, A344964.

%K nonn,cons

%O 1,3

%A _Jwalin Bhatt_, Oct 17 2025