A390737 Number of decimal places which are known after using n terms of A084580 and treating them as the coefficients of a continued fraction.

0, 0, 1, 1, 2, 2, 4, 4, 5, 5, 5, 7, 7, 8, 9, 9, 10, 11, 12, 12, 12, 14, 14, 15, 17, 17, 18, 19, 19, 20, 20, 21, 22, 22, 22, 22, 25, 25, 27, 27, 28, 29, 29, 30, 30, 32, 33, 33, 34, 34, 35, 36, 36, 37, 38, 39, 40, 42, 42, 43, 44, 44, 44, 46, 46, 47, 47, 47, 50

Offset

0,5

Comments

The digits produced by the continued fraction are that of A372869.

Let c(n) be the n-th convergent resulting from A084580.

Further continued fraction terms are taken to be unknown and the worst case (maximum absolute difference) is a single additional 1 and so at worst c(n+1).

a(n) is therefore the length of agreement between c(n) and c(n+1).

Limit_{n->oo} n/a(n) seems to approach a value between 1/(2*log_10(Pi)) and Loch's constant (A086702). - Corrected by Jwalin Bhatt, Jun 29 2026

Links

Jwalin Bhatt, Table of n, a(n) for n = 0..5000

Wikipedia, Lochs's theorem.

Example

For n=6, the continued fraction of 6 terms is c(n) = [0; 1,1,2,1,1,3] = 25/43 and the worst case continuation is c(n+1) = 32/55,
    25/43 = 0.581 39...
    32/55 = 0.581 81...
            ^ ^^^ a(7) = 4 places agree

Prog

(Python)  # Using sample_gauss_kuzmin_distribution function from A084580.
from sympy import floor, continued_fraction_convergents
from collections import deque
from os.path import commonprefix
def reliable_digits_from_cf(coeffs, prec):
    bound1, bound2 = deque(continued_fraction_convergents(coeffs+[1]), maxlen=2)
    order = 10**prec
    trunc_bound1 = floor(bound1*order) / order
    trunc_bound2 = floor(bound2*order) / order
    rel_digits = commonprefix([f'{trunc_bound1:.{prec}f}', f'{trunc_bound2:.{prec}f}'])
    return len(rel_digits) - 1 if rel_digits else 0
coeffs = sample_gauss_kuzmin_distribution(100)
A390737 = [reliable_digits_from_cf([0]+coeffs[:i], 1+int(i*1.031)) for i in range(len(coeffs)+1)]

Crossrefs

Cf. A062542, A084580, A086819, A372869.

Sequence in context: A347660 A116648 A152850 * A036714 A319398 A260734

Adjacent sequences: A390734 A390735 A390736 * A390738 A390739 A390740

Keyword

nonn,base

Author

Jwalin Bhatt, Nov 16 2025

Extensions

Edited and corrected by Jwalin Bhatt, Feb 04 2026

Status

approved