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%I #40 Jan 31 2026 16:57:09
%S 1,2,12,12,120,120,2520,2520,5040,5040,55440,55440,720720,720720,
%T 720720,720720,24504480,24504480,1396755360,1396755360,6983776800,
%U 6983776800,160626866400,160626866400,160626866400,160626866400,160626866400,160626866400
%N a(n) = the least common multiple of all natural numbers k such that phi(k) <= n.
%C phi(n) = A000010(n). The range of the phi function is A002202; the latter excludes odd numbers > 1 and some even numbers as well, and explains why some terms in this sequence are repeated. The (multi-valued) inverse of the phi function is A032447. Note that in computing the least common multiple for the present sequence, only prime power values of k are required, which makes the search shorter.
%C All terms of this sequence are the least having that prime signature (A025487).
%C It appears that log(a(n))/n approaches 1 as n increases without bound.
%H Andrew Howroyd, <a href="/A392539/b392539.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Product_{k=1..n} A381887(k).
%e a(4) = the least common multiple of all natural numbers k such that phi(k) <= 4. The full list of such k is {1, 2, 3, 4, 5, 6, 8, 10, 12}, and the least common multiple of that set is 120.
%t seq[len_] := With[{phi = EulerPhi[Range[If[len <= 2, 3*len, len^2]]]}, Table[LCM @@ Flatten[Position[phi, _?(# <= n &)]], {n, 1, len}]]; seq[30] (* _Amiram Eldar_, Jan 16 2026 *)
%Y Cf. A000010, A002202, A007614, A025487, A032447, A378636, A378638, A381887.
%K nonn
%O 0,2
%A _Hal M. Switkay_, Jan 15 2026