A338140 a(n) is the smallest number with n refactorable divisors.

1, 2, 8, 18, 24, 36, 108, 180, 72, 216, 288, 1944, 360, 1080, 1920, 720, 1800, 2160, 5400, 1440, 6720, 3600, 12600, 4320, 16200, 5760, 12960, 38016, 13440, 45360, 35280, 10080, 21600, 28800, 67200, 51840, 215040, 20160, 30240, 97200, 50400, 64800, 144000

Offset: 1

Author

Jaroslav Krizek, Oct 24 2020

Keywords

• nonn

Comments

a(n) is the greedy inverse of A336041: the smallest number with exactly n divisors d such that d / tau(d) is also an integer.

Numbers 1 and 2 are only numbers m such that d / tau(d) is an integer for all divisors d of m.

Examples

			a(3) = 8 because 8 with divisors 1, 2, 4 and 8 is the smallest number with 3 refactorable divisors: 1 / tau(1) = 1, 2 / tau(2) = 1, 8 / tau(8) = 2.
		

Crossrefs

Cf. A336041, A033950 (refactorable numbers).

Cf. A335182, A335665.

Programs

Magma

[Min([m: m in[1..10^5] | #[d: d in Divisors(m) | IsIntegral(d / #Divisors(d))] eq n]): n in [1..12]]

Mathematica

f[n_] := DivisorSum[n, 1 &, Divisible[#, DivisorSigma[0, #]] &]; m = 43; s = Table[0, {m}]; c = 0; n = 1; While[c < m, i = f[n]; If[i <= m && s[[i]] == 0, c++; s[[i]] = n]; n++]; s (* Amiram Eldar, Oct 24 2020 *)

Formula

a(n) = min{ k: A336041(k)=n}. - R. J. Mathar, Nov 24 2020