A389850 Decimal expansion of the absolute value of the sum of gamma function at all its extremas.

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

Offset

1,3

Comments

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

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

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.

Links

Table of n, a(n) for n=1..88.

Wikipedia, Digamma Roots.

Wikipedia, Gamma Extrema.

Formula

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

Example

abs(-1.04433149368802847521089...).
The sum begins 0.885603... + (-3.544643) + 2.302407 + ... (being A030171 at x_0=A030169, then Gamma(A175472), Gamma(A175473), ...).

Crossrefs

Cf. A030171, A175472, A175473, A344964.

Sequence in context: A364128 A120886 A228777 * A329205 A023402 A073261

Adjacent sequences: A389847 A389848 A389849 * A389851 A389852 A389853

Keyword

nonn,cons

Author

Jwalin Bhatt, Oct 17 2025

Status

approved