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

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, 27, 32, 32, 30, 32, 32, 64, 36, 36, 36, 48, 38, 40, 40, 48, 42, 48, 44, 48, 48, 48, 48, 64, 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

Offset: 1

Author

Dan Dart, Jul 03 2022

Keywords

• nonn

Comments

Distinct from 2^A073093 because of the proviso that a(n) > n and bigomega(a(n)) > bigomega(n).

Examples

			For n = 1, a(1) = 2, since 2 is the first number satisfying 2 > 1 and bigomega(2) = 1 > bigomega(1) = 0.
For n = 5, a(5) = 8, since 8 is the first number satisfying 8 > 5 and bigomega(8) = 3 > bigomega(5) = 1.
For n = 12, a(12) = 16, since 16 is the first number satisfying 16 > 12 and bigomega(16) = 4 > bigomega(12) = 3.
		

Crossrefs

Cf. A073093, A001222.

Programs

Haskell

import Data.Numbers.Primes
result :: [Integer]
result = fmap (
  \n -> head (
      dropWhile (
          \m -> length (primeFactors m :: [Integer]) <= length (primeFactors n :: [Integer])
      )
      [n..]
  )
  ) [1..]

Maple

A355467 := proc(n)
    local a,nOmega ;
    nOmega := A001222(n) ;
    for a from n+1 do
        if A001222(a) > nOmega then
            return a;
        end if;
    end do;
end proc:
seq(A355467(n),n=1..80) ; # R. J. Mathar, May 05 2023

PARI

a(n) = my(k=n+1, nb=bigomega(n)); while (bigomega(k) <= nb, k++); k; \\ Michel Marcus, Jul 05 2022

Formula

a(2^n) = 2^(n+1) because the smallest extra factor is 2.

a(3*2^n) = 2^(n+2) because 4 (i.e., 2^2) is the next biggest pair of factors.