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A355467 a(n) is the smallest number which is greater than n and has more prime factors (with multiplicity) than n.

%I A355467 #22 May 05 2023 07:57:18
%S A355467 2,4,4,8,6,8,8,16,12,12,12,16,14,16,16,32,18,24,20,24,24,24,24,32,27,
%T A355467 27,32,32,30,32,32,64,36,36,36,48,38,40,40,48,42,48,44,48,48,48,48,64,
%U A355467 50,54,52,54,54,64,56,64,60,60,60,64,62,63,64,128,66,72,68,72,70,72,72,96,74,75,80,80,78,80,80,96,96
%N A355467 a(n) is the smallest number which is greater than n and has more prime factors (with multiplicity) than n.
%C A355467 Distinct from 2^A073093 because of the proviso that a(n) > n and bigomega(a(n)) > bigomega(n).
%F A355467 a(2^n) = 2^(n+1) because the smallest extra factor is 2.
%F A355467 a(3*2^n) = 2^(n+2) because 4 (i.e., 2^2) is the next biggest pair of factors.
%e A355467 For n = 1, a(1) = 2, since 2 is the first number satisfying 2 > 1 and bigomega(2) = 1 > bigomega(1) = 0.
%e A355467 For n = 5, a(5) = 8, since 8 is the first number satisfying 8 > 5 and bigomega(8) = 3 > bigomega(5) = 1.
%e A355467 For n = 12, a(12) = 16, since 16 is the first number satisfying 16 > 12 and bigomega(16) = 4 > bigomega(12) = 3.
%p A355467 A355467 := proc(n)
%p A355467     local a,nOmega ;
%p A355467     nOmega := A001222(n) ;
%p A355467     for a from n+1 do
%p A355467         if A001222(a) > nOmega then
%p A355467             return a;
%p A355467         end if;
%p A355467     end do;
%p A355467 end proc:
%p A355467 seq(A355467(n),n=1..80) ; # _R. J. Mathar_, May 05 2023
%o A355467 (Haskell)
%o A355467 import Data.Numbers.Primes
%o A355467 result :: [Integer]
%o A355467 result = fmap (
%o A355467   \n -> head (
%o A355467       dropWhile (
%o A355467           \m -> length (primeFactors m :: [Integer]) <= length (primeFactors n :: [Integer])
%o A355467       )
%o A355467       [n..]
%o A355467   )
%o A355467   ) [1..]
%o A355467 (PARI) a(n) = my(k=n+1, nb=bigomega(n)); while (bigomega(k) <= nb, k++); k; \\ _Michel Marcus_, Jul 05 2022
%Y A355467 Cf. A073093, A001222.
%K A355467 nonn
%O A355467 1,1
%A A355467 _Dan Dart_, Jul 03 2022