A130860 Number of decimal places of Pi given by integer approximations of the form a^(1/n).

0, 0, 2, 2, 4, 2, 3, 4, 5, 5, 6, 6, 6, 6, 9, 9, 9, 10, 10, 11, 11, 13, 12, 13, 13, 14, 14, 14

Offset: 1

Author

Stephen McInerney (spmcinerney(AT)hotmail.com), Jul 22 2007

Keywords

• nonn
• base
• more

Comments

Approximations are rounded, not truncated; see the example for n=2. Note that this can produce anomalous results; e.g., 0.148 does not match 0.152 to 1-place accuracy, but does match it to 2-place accuracy. - Franklin T. Adams-Watters, Mar 29 2014

Examples

			a(8)=4 because 9489^(1/8) = 3.1416... is Pi accurate to 4 decimal places.
a(2)=0. 10^(1/2) = 3.16... rounded to one place is 3.2, while Pi to one place is 3.1.
		

Crossrefs

Cf. A002160.

Programs

Python

from math import pi, floor, ceil
def round(x):
    return math.floor(x + 0.5)
def decimal_places(x, y):
    dp = -1
    # Compare integer part, shift 1 dp
    while floor(x + 0.5) == floor(y + 0.5) and x and y:
        x = (x - floor(x)) * 10
        y = (y - floor(y)) * 10
        dp = dp + 1
    return dp
for n in range(1, 30):
    pi_to_the_n = pow(pi, n)
    pi_to_the_n_rnd = round(pi_to_the_n)
    pi_approx = pow(pi_to_the_n_rnd, 1.0 / n)
    dps = decimal_places(pi_approx, pi)
    print(dps)

Formula

a(n) is the number of decimal_places in (round(Pi^n))^1/n w.r.t. Pi.

Note that round(Pi^n) is the sequence A002160 (Nearest integer to Pi^n).