

A061799


Smallest number with at least n divisors.


13



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
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OFFSET

1,2


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


LINKS

T. D. Noe, Table of n, a(n) for n=1..2000 (using A002182)


EXAMPLE

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).


MATHEMATICA

Reap[ For[ n = 1, n <= 100, n++, s = n; While[ DivisorSigma[0, s] < n, s++]; Sow[s] ] ][[2, 1]] (* JeanFrançois 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 *)


PROG

(PARI) for(n=1, 100, s=n; while(numdiv(s)<n, s++); print1(s, ", "))
(Haskell)
import Data.List (findIndex)
import Data.Maybe (fromJust)
a061799 n = succ $ fromJust $ findIndex (n <=) $ map a000005 [1..]
 Reinhard Zumkeller, Apr 01 2011


CROSSREFS

Cf. A000005, A002182, A002183, A005179, A213918.
Sequence in context: A053146 A056675 A062857 * A076868 A056793 A137387
Adjacent sequences: A061796 A061797 A061798 * A061800 A061801 A061802


KEYWORD

nice,nonn


AUTHOR

Henry Bottomley, Jun 22 2001


EXTENSIONS

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


STATUS

approved



