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a(n) minimizes (over the integers) the absolute difference between Pi and x(n) + 1/a(n), where x(n) is Pi truncated at the n-th decimal digit.
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%I #8 May 13 2017 23:46:03

%S 7,24,628,1687,10793,376848,1530012,18660270,278567575,1695509434,

%T 11136696004,102111268282,1260654956982,10725187563686,

%U 308788493220130,4193528956200936,25999253094360135,118166387818704585,2161492060929047665,15963377896404315144

%N a(n) minimizes (over the integers) the absolute difference between Pi and x(n) + 1/a(n), where x(n) is Pi truncated at the n-th decimal digit.

%e 3 + 1/7 is closest to Pi in absolute value among numbers of the form 3 + 1/k (k an integer); 3.1 + 1/24 is closest to Pi in absolute value among numbers of the form 3.1 + 1/k (k an integer); 3.14 + 1/628 is closest to Pi in absolute value among numbers of the form 3.14 + 1/k (k an integer).

%t Table[

%t truncpi = Floor[10^(n - 1)*Pi]/10^(n - 1);

%t SortBy[

%t {Floor[1/(Pi - truncpi)], Ceiling[1/(Pi - truncpi)]},

%t N[Abs[Pi - (truncpi + 1/#)]] &

%t ][[1]],

%t {n, 1, 20}] (* first 20 terms *)

%Y Cf. A074783.

%K nonn,base

%O 1,1

%A _Jason Zimba_, May 13 2017