A118476 - OEIS

%I #16 Feb 16 2025 08:33:01

%S 1,2,6,20,81,408,2480,17376,139040,1251450,12514816,137663064,

%T 1651956992,21475443200,300656206080,4509843098112,72157489576704,

%U 1226677322842112,22080191811166208,419523644412176256,8390472888243683328,176199930653117513728

%N a(0) = 1; a(n) is least k with n prime factors and k > n*a(n-1).

%C This is a super-polynomial function, as for positive n, a(n) > n!.

%C Prime factors counted with multiplicity. - _Harvey P. Dale_, Aug 25 2019

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>.

%F a(0) = 1; a(n) least n-almost prime > n*a(n-1).

%e a(1) = 2 because 2 is the smallest prime (integer with 1 prime factor) greater than 1 * 1 = 1.

%e a(2) = 6 because 6 = 2 * 3 is the smallest semiprime (integer with 2 prime factors) greater than 2 * 2 = 4.

%e a(3) = 20 because 20 = 2^2*5 is the smallest 3-almost prime (integer with 3 prime factors) greater than 3 * 6 = 18.

%p A118476 := proc(n) option remember; local k; if n = 0 then 1; else for k from n*procname(n-1)+1 do if numtheory[bigomega](k) = n then return k; end if; end do: end if; end proc:

%p seq(A118476(n),n=0..14) ; # _R. J. Mathar_, Dec 22 2010

%t lkpf[{n_,a_}]:=Module[{k=a(n+1)+1},While[PrimeOmega[k]!=n+1,k++];{n+1,k}]; NestList[lkpf,{0,1},21][[All,2]] (* _Harvey P. Dale_, Aug 25 2019 *)

%Y Cf. A000040, A001358, A055496, A076656.

%K easy,nonn,changed

%O 0,2

%A _Jonathan Vos Post_, May 04 2006

%E Terms corrected from a(4) on by _R. J. Mathar_, Dec 22 2010

%E a(15)-a(21) from _Donovan Johnson_, Jan 06 2011