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

1,2

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.

Links

Table of n, a(n) for n=1..43.

Formula

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A336041, A033950 (refactorable numbers).

Cf. A335182, A335665.

Sequence in context: A117612 A320662 A171613 * A306394 A109136 A295522

Adjacent sequences: A338137 A338138 A338139 * A338141 A338142 A338143

Keyword

nonn

Author

Jaroslav Krizek, Oct 24 2020

Status

approved