A073093 -id:A073093 - cp's OEIS frontend

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

Dan Dart, Jul 03 2022

nonn

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

			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.

Cf. A073093, A001222.

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

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

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

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.

A356855 a(n) is the least number m such that u defined by u(i) = bigomega(m + 2i) satisfies u(i) = u(0) for 0 <= i < n and u(n) != u(0), or -1 if no such number exists.

Original entry on oeis.org

1, 4, 3, 215, 213, 1383, 3091, 8129, 151403, 151401, 2560187, 33396293, 33396291, 56735777, 1156217487, 2514196079

Offset: 1

Jean-Marc Rebert, Sep 04 2022

			Let u be defined by u(i) = bigomega(3 + 2i). u(i) = 1 for 0 <= i < 3 and u(3) = 2 != 1, and 3 is the smallest such number, hence a(3) = 3.
Let u be defined by u(i) = bigomega(4 + 2i). u(i) = 2 for 0 <= i < 2 and u(3) = 3 != 2 , and 4 is the smallest such number, hence a(2) = 4.
Let u be defined by u(i) = bigomega(151403 + 2i). u(i) = 3 for 0 <= i < 9 and u(9) = 2 != 3, and 151403 is the smallest such number, hence a(9) = 151403.

Cf. A113752, A356893.

Cf. A073093 and A091304 (the 2 bisections of A001222).

u(m,i)=bigomega(m+2*i)
card(m)=my(k=u(m,0),c=0);while(u(m,c)==k,c++);c
a(n)=my(c=0);for(m=1,+oo,c=card(m);if(c==n,return(m)))

cp's OEIS Frontend

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

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