OFFSET
1,3
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
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).
EXAMPLE
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.
PROG
(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)
CROSSREFS
KEYWORD
nonn,base,more
AUTHOR
Stephen McInerney (spmcinerney(AT)hotmail.com), Jul 22 2007
STATUS
approved