A381898 Decimal expansion of exp(Sum_{k>=2} log_2(k)/(k * 2^k)).

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

Offset: 1

Author

Jwalin Bhatt, Mar 09 2025

Keywords

• nonn
• cons

Comments

The geometric mean of the logarithmic distribution with parameter value 1/2 (A381900) approaches this constant. In general, for parameter value p, it approaches exp(PolyLog'(1,p)/log(1-p)).

Examples

			1.2833031711874068193927988...
		

Crossrefs

Cf. A381456, A381900.

Programs

Mathematica

N[Exp [Sum[Log2[i]/(i*2^i), {i, 2, Infinity}]], 120]
N[Exp[-Derivative[1, 0][PolyLog][1, 1/2]/Log[2]], 120]

Python

from mpmath import polylog, diff, exp, log, mp
mp.dps = 120
const = exp(-diff(lambda n: polylog(n, 1/2), 1)/log(2))
A381898 = [int(d) for d in mp.nstr(const, n=mp.dps)[:-1] if d != '.']

Formula

Equals exp(-PolyLog'(1,1/2)/log(2)), where PolyLog'(x,y) represents the derivative of the polylogarithm w.r.t. x.

From Jwalin Bhatt, May 07 2025: (Start)

Equals exp(Sum_{k>=2} log_2(k)/(k*2^k)).

Equals (Product_{k>=2} k^(1/(k*2^k))) ^ (1/log(2)). (End)

Extensions

a(120) corrected by Sean A. Irvine, Apr 14 2025