A061707 Smallest number with exactly n^2 divisors.

1, 6, 36, 120, 1296, 1260, 46656, 7560, 44100, 45360, 60466176, 110880, 2176782336, 1632960, 1587600, 1081080, 2821109907456, 3880800, 101559956668416, 6486480, 57153600, 2116316160, 131621703842267136, 21621600, 1944810000, 76187381760

Offset: 1

Author

Labos Elemer, Jun 19 2001

Keywords

• nonn

Comments

For n = prime see A061234.

Examples

			For n = 6: a(6) = 1260 = 2*2*3*3*5*7 and d(1260) = 36.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1288 (terms 1..44 from Amiram Eldar, terms 45..330 from David A. Corneth)

Crossrefs

Cf. A000005, A000040, A000290, A005179, A061234, A300357 (1st bisection).

Programs

Mathematica

a[n_]:=(f=Flatten[Table@@@FactorInteger[n^2]]; Min[Times@@(Prime@Range@Length[s=Reverse@#-1]^s)&/@Union[Flatten[(ff=#;Union[Sort/@(Times@@@TakeList[ff,#]&/@Flatten[Permutations/@IntegerPartitions[Length@f],1])])&/@Permutations@f,1]]]);Array[a,26] (* Giorgos Kalogeropoulos, Dec 10 2021 *)

PARI

a(n) = {my(k = 1); while (numdiv(k) != n^2, k++); k;} \\ Michel Marcus, Sep 05 2017

Formula

a(n) = Min_{x : d(x) = n^2} = A005179(A000290(n)).

A000005(a(n)) = A000290(n).

a(prime(n)) = A061234(n). - Bernard Schott, Jun 21 2019

a(n) = A005179(n^2). - Amiram Eldar, Feb 04 2025

Extensions

More terms from Don Reble, Nov 08 2001

Offset corrected by Hagen von Eitzen, Jun 05 2009

a(24)-a(26) from Ray Chandler, Jun 19 2009