A061799 - OEIS

%I #43 Jan 01 2022 09:55:28

%S 1,2,4,6,12,12,24,24,36,48,60,60,120,120,120,120,180,180,240,240,360,

%T 360,360,360,720,720,720,720,720,720,840,840,1260,1260,1260,1260,1680,

%U 1680,1680,1680,2520,2520,2520,2520,2520,2520,2520,2520,5040,5040,5040

%N Smallest number with at least n divisors.

%C Smallest number which can be expressed as the least common multiple of n distinct numbers. - _Amarnath Murthy_, Nov 27 2002

%C 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

%H Amiram Eldar, <a href="/A061799/b061799.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..2000 from T. D. Noe)

%e 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).

%t 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 *)

%t 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 *)

%o (PARI) for(n=1,100,s=n; while(numdiv(s)<n,s++); print1(s,","))

%o (Haskell)

%o import Data.List (findIndex)

%o import Data.Maybe (fromJust)

%o a061799 n = succ $ fromJust $ findIndex (n <=) $ map a000005 [1..]

%o -- _Reinhard Zumkeller_, Apr 01 2011

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

%K nice,nonn

%O 1,2

%A _Henry Bottomley_, Jun 22 2001

%E Replaced "factors" by "divisors" in definition and example _M. F. Hasler_, Oct 24 2010