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

1,1

Comments

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

Links

Table of n, a(n) for n=1..81.

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.

Example

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.

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

Prog

(Haskell)
import Data.Numbers.Primes
result :: [Integer]
result = fmap (
  \n -> head (
      dropWhile (
          \m -> length (primeFactors m :: [Integer]) <= length (primeFactors n :: [Integer])
      )
      [n..]
  )
  ) [1..]
(PARI) a(n) = my(k=n+1, nb=bigomega(n)); while (bigomega(k) <= nb, k++); k; \\ Michel Marcus, Jul 05 2022

Crossrefs

Cf. A073093, A001222.

Sequence in context: A005884 A229913 A285326 * A079890 A337973 A223592

Adjacent sequences: A355464 A355465 A355466 * A355468 A355469 A355470

Keyword

nonn

Author

Dan Dart, Jul 03 2022

Status

approved