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A342831 a(n) is the smallest positive integer k such that the n-dimensional cube [0,k]^n contains at least as many internal lattice points as external lattice points.

%I A342831 #19 May 30 2021 21:01:46
%S A342831 3,6,9,12,15,18,21,24,26,29,32,35,38,41,44,47,50,52,55,58,61,64,67,70,
%T A342831 73,76,78,81,84,87,90,93,96,99,101,104,107,110,113,116,119,122,125,
%U A342831 127,130,133,136,139,142,145,148,151,153,156,159,162,165,168,171,174,177,179,182
%N A342831 a(n) is the smallest positive integer k such that the n-dimensional cube [0,k]^n contains at least as many internal lattice points as external lattice points.
%H A342831 Alois P. Heinz, <a href="/A342831/b342831.txt">Table of n, a(n) for n = 1..10000</a>
%F A342831 a(1) = 3 and a(n) = floor(2^(1/n+1)/(2^(1/n)-1)) for n > 1.
%F A342831 a(n) = A078608(n) + 1.
%e A342831 a(2) > 5 because the number of internal lattice points = 4^2 = 16 < 20 = 6^2 - 16 = the number of external lattice points, therefore a(2)=6 because the number of internal lattice points = 5^2 = 25 > 24 = 7^2 - 25 = number of external lattice points.
%p A342831 a:= n-> ceil(1+2/(2^(1/n)-1)):
%p A342831 seq(a(n), n=1..65);  # _Alois P. Heinz_, Apr 20 2021
%t A342831 a[1] = 3; a[n_] := Floor[2^(1/n + 1)/(2^(1/n) - 1)]; Array[a, 100] (* _Amiram Eldar_, Mar 31 2021 *)
%Y A342831 Cf. A078608.
%K A342831 nonn
%O A342831 1,1
%A A342831 _Gary Yane_, Mar 23 2021