A061799 Smallest number with at least n divisors.

1, 2, 4, 6, 12, 12, 24, 24, 36, 48, 60, 60, 120, 120, 120, 120, 180, 180, 240, 240, 360, 360, 360, 360, 720, 720, 720, 720, 720, 720, 840, 840, 1260, 1260, 1260, 1260, 1680, 1680, 1680, 1680, 2520, 2520, 2520, 2520, 2520, 2520, 2520, 2520, 5040, 5040, 5040

Offset: 1

Author

Henry Bottomley, Jun 22 2001

Keywords

• nice
• nonn

Comments

Smallest number which can be expressed as the least common multiple of n distinct numbers. - Amarnath Murthy, Nov 27 2002

Also smallest possible member of a set of n+1 numbers with pairwise distinct GCD's. [Following an observation by Charles R Greathouse IV] (Proof: If the smallest number min(S) of the set (with card(S)=n+1) has a distinct GCD with each of the other n numbers, then it must have at least n distinct divisors (because any GCD is a divisor). It is then easy to choose larger members of the set so that all pairs of elements have pairwise distinct GCD's, e.g., by successively multiplying by distinct and sufficiently large primes.) - M. F. Hasler, Mar 05 2013

Examples

			a(5)=12 since every number less than 12 has fewer than five divisors (1 has one; 2,3,5,7 and 11 have two each; 4 and 9 have three each; 6,8 and 10 have four each) while 12 has at least five (in fact it has six: 1,2,3,4,6 and 12).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from T. D. Noe)

Crossrefs

Cf. A000005, A002182, A002183, A005179, A213918.

Programs

Haskell

import Data.List (findIndex)
import Data.Maybe (fromJust)
a061799 n = succ $ fromJust $ findIndex (n <=) $ map a000005 [1..]
-- Reinhard Zumkeller, Apr 01 2011

Mathematica

Reap[ For[ n = 1, n <= 100, n++, s = n; While[ DivisorSigma[0, s] < n, s++]; Sow[s] ] ][[2, 1]] (* Jean-François Alcover, Feb 16 2012, after Pari *)
With[{ds=Table[{n,DivisorSigma[0,n]},{n,6000}]},Table[SelectFirst[ds,#[[2]] >= k&],{k,60}]][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 15 2019 *)

PARI

for(n=1,100,s=n; while(numdiv(s)
				

Extensions

Replaced "factors" by "divisors" in definition and example M. F. Hasler, Oct 24 2010