A380373 Decimal expansion of Sum_{i>=1} 1/2^A082851(i).

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

Offset: 0

Author

Jwalin Bhatt, Jan 23 2025

Keywords

• nonn
• cons

Comments

This number has the property that the geometric mean of the differences (A082850) in the positions of 1s (A082851) of its binary representation (A380372) approaches the Somos constant (A112302).

Examples

			0.8641913214950458_10 -> 0.1101110100111011101_2 (A380372)
Positions of 1s -> 1,2,4,5,6,8,11,12,13,15,... (A082851)
Difference in positions of 1s -> 1,1,2,1,1,2,3,1,1,2,1,1,2,3,4,... (A082850)
Geomtric Mean -> 1.66168794963359... (A112302)
		

Links

• Wikipedia, Somos Constant

Crossrefs

Cf. A082850, A082851, A380372 (binary expansion).

Programs

Python

from itertools import count, islice
from fractions import Fraction
import os
def A380372_gen():
    S = []
    for n in count(1):
        yield from (m:=S+[0]*(n-1)+[1])
        S += m
def bin_to_frac_interval(binary_repr):
    lower_bound, last_bit_id = 0, 0
    for i, bit in enumerate(binary_repr, start=1):
        if bit:
            lower_bound += Fraction(1, 2**i)
            last_bit_id = i
    upper_bound = lower_bound + Fraction(1, 2**last_bit_id)
    return lower_bound, upper_bound
n_binary_terms = 400
diff_bin = islice(A380372_gen(), n_binary_terms)
lower, upper = bin_to_frac_interval(diff_bin)
lower, upper = str(int(lower*10**(n_binary_terms//3))), str(int(upper*10**(n_binary_terms//3)))
A380373 = os.path.commonprefix([lower, upper])

Formula

Equals Sum_{i>=1} 1/2^A082851(i).

Equals Sum_{i>=1} A380372(i)/2^i.