A061283 Smallest number with exactly 2n-1 divisors.

1, 4, 16, 64, 36, 1024, 4096, 144, 65536, 262144, 576, 4194304, 1296, 900, 268435456, 1073741824, 9216, 5184, 68719476736, 36864, 1099511627776, 4398046511104, 3600, 70368744177664, 46656, 589824, 4503599627370496, 82944

Offset: 1

Author

Labos Elemer, May 22 2001

Keywords

• nonn

Comments

The terms are always squares (because the divisors of a nonsquare N come in pairs, d and N/d, and so their number is always even - N. J. A. Sloane, Dec 26 2018).

Examples

			For n=15, a(15)=144 with 15 divisors: 1,2,3,4,6,8,9,12,16,18,24,36,48,72 and 144.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1661 (terms 1..1000 from T. D. Noe)

Crossrefs

Cf. A000005, A000290, A005408, A003680, A016017, A037992, A055079, A048691.

Second bisection of A005179.

Programs

Mathematica

mp[1, m_] := {{}}; mp[n_, 1] := {{}}; mp[n_?PrimeQ, m_] := If[m < n, {}, {{n}}]; mp[n_, m_] := Join @@ Table[Map[Prepend[#, d] &, mp[n/d, d]], {d, Select[Rest[Divisors[n]], # <= m &]}]; mp[n_] := mp[n, n]; Table[mulpar = mp[2*n-1] - 1; Min[Table[Product[Prime[s]^mulpar[[j, s]], {s, 1, Length[mulpar[[j]]]}], {j, 1, Length[mulpar]}]], {n, 1, 100}] (* Vaclav Kotesovec, Apr 04 2021 *)

Formula

a(n) = Min{k | A000005(k)=2n-1}.

a((p+1)/2) = 2^(p-1) for odd prime p. [Corrected by Jianing Song, Aug 30 2021]

From Jianing Song, Aug 30 2021: (Start)

a(n) = A016017(n)^2.

a(n) <= 2^(2n-2), where the equality holds if and only if n=1 or 2n-1 is prime. (End)