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%I #48 Apr 07 2015 15:36:37
%S 1,2,102,249,294,1635,3647,5105,6954,357593,416507,497533,821581,
%T 1299504,1457054,39082410,45969853,47607985,86389357,121338042,
%U 183331620,191525092,198003319,388055713,427238910,570434345,678096433
%N Numbers obtained by removing the first a(n) > 0 decimal digits of Pi set new records for closeness to Pi.
%C Suppose you forget the first a(n) digits of Pi after the 3, but remember the rest - these values are the record-setting best approximations to Pi.
%C a(28) > 4.9*10^9. - _Robert G. Wilson v_, Apr 03 2015
%e a(1) = 1, giving an approximation 3.4159...
%e a(2) = 2, because 3.1592... is closer to Pi than 3.4159...
%e a(3) = 102, because 3.14808... is closer to Pi than any value obtained by removing fewer than 102 of the first decimal digits of Pi.
%e a(27) = 678096433, because 3.141592653607137185825... is closer to Pi than any value obtained by removing fewer than 678096433 of the first decimal digits of Pi. - _Robert G. Wilson v_, Apr 04 2015
%t pi = N[Pi - 3, 1000000]; k = 1; d = Infinity; lst = {}; While[k < 990000, pi = 10 pi - IntegerPart[10 pi]; If[ Abs[Pi - 3 - pi] < d, d = Abs[Pi - 3 - pi]; AppendTo[lst, k]; Print[k]]; k++]; lst (* _Robert G. Wilson v_, Apr 01 2015 *)
%o (Python)
%o def a256516():
%o ....best = 1
%o ....yield 1
%o ....i = 2
%o ....while True:
%o ........if i>=len(pi):
%o ............return
%o ........a = pi[i]
%o ........valid = True
%o ........o = 1
%o ........while valid:
%o ............pi_approx = int(pi[:o])
%o ............a_approx = abs(int(pi[i:i+o])-pi_approx)
%o ............b_approx = abs(int(pi[best:best+o])-pi_approx)
%o ............if abs(b_approx-a_approx)>10:
%o ................valid = False
%o ............else:
%o ................o+=1
%o ........if a_approx<b_approx:
%o ............best = i
%o ............yield i
%o ........i+=1
%Y Cf. Decimal expansion of Pi: A000796.
%K nonn,base,more
%O 1,2
%A _Christian Perfect_, Apr 01 2015
%E a(10)-a(13) from _Robert G. Wilson v_, Apr 01 2015
%E a(14)-a(19) from _Robert G. Wilson v_, Apr 02 2015
%E a(20)-a(27) from _Robert G. Wilson v_, Apr 03 2015
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