A392539 a(n) = the least common multiple of all natural numbers k such that phi(k) <= n.

1, 2, 12, 12, 120, 120, 2520, 2520, 5040, 5040, 55440, 55440, 720720, 720720, 720720, 720720, 24504480, 24504480, 1396755360, 1396755360, 6983776800, 6983776800, 160626866400, 160626866400, 160626866400, 160626866400, 160626866400, 160626866400

Offset

0,2

Comments

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.

All terms of this sequence are the least having that prime signature (A025487).

It appears that log(a(n))/n approaches 1 as n increases without bound.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

a(n) = Product_{k=1..n} A381887(k).

Example

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.

Mathematica

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 *)

Crossrefs

Cf. A000010, A002202, A007614, A025487, A032447, A378636, A378638, A381887.

Sequence in context: A198532 A198477 A075178 * A145618 A306693 A169900

Adjacent sequences: A392536 A392537 A392538 * A392540 A392541 A392542

Keyword

nonn

Author

Hal M. Switkay, Jan 15 2026

Status

approved