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%I #24 Jan 27 2022 21:59:33
%S 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
%N Number of decimal places of Pi given by integer approximations of the form a^(1/n).
%C 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
%F a(n) is the number of decimal_places in (round(Pi^n))^1/n w.r.t. Pi.
%F Note that round(Pi^n) is the sequence A002160 (Nearest integer to Pi^n).
%e a(8)=4 because 9489^(1/8) = 3.1416... is Pi accurate to 4 decimal places.
%e a(2)=0. 10^(1/2) = 3.16... rounded to one place is 3.2, while Pi to one place is 3.1.
%o (Python)
%o from math import pi, floor, ceil
%o def round(x):
%o return math.floor(x + 0.5)
%o def decimal_places(x, y):
%o dp = -1
%o # Compare integer part, shift 1 dp
%o while floor(x + 0.5) == floor(y + 0.5) and x and y:
%o x = (x - floor(x)) * 10
%o y = (y - floor(y)) * 10
%o dp = dp + 1
%o return dp
%o for n in range(1, 30):
%o pi_to_the_n = pow(pi, n)
%o pi_to_the_n_rnd = round(pi_to_the_n)
%o pi_approx = pow(pi_to_the_n_rnd, 1.0 / n)
%o dps = decimal_places(pi_approx, pi)
%o print(dps)
%Y Cf. A002160.
%K nonn,base,more
%O 1,3
%A Stephen McInerney (spmcinerney(AT)hotmail.com), Jul 22 2007
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