A080257 Numbers having at least two distinct or a total of at least three prime factors.

6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100

Offset: 1

Author

Reinhard Zumkeller, Feb 10 2003

Keywords

• nonn
• easy

Comments

Complement of A000430; A080256(a(n)) > 3.

A084114(a(n)) > 0, see also A084110.

Also numbers greater than the square of their smallest prime-factor: a(n)>A020639(a(n))^2=A088377(a(n));

a(n)>A000430(k) for n<=13, a(n) < A000430(k) for n>13.

Numbers with at least 4 divisors. - Franklin T. Adams-Watters, Jul 28 2006

Union of A024619 and A033942; A211110(a(n)) > 2. - Reinhard Zumkeller, Apr 02 2012

Also numbers > 1 that are neither prime nor a square of a prime. Also numbers whose omega-sequence (A323023) has sum > 3. Numbers with omega-sequence summing to m are: A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7). - Gus Wiseman, Jul 03 2019

Numbers n such that sigma_2(n)*tau(n) = A001157(n)*A000005(n) >= 4*n^2. Note that sigma_2(n)*tau(n) >= sigma(n)^2 = A072861 for all n. - Joshua Zelinsky, Jan 23 2025

Examples

			8=2*2*2 and 10=2*5 are terms; 4=2*2 is not a term.
From _Gus Wiseman_, Jul 03 2019: (Start)
The sequence of terms together with their prime indices begins:
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  27: {2,2,2}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
(End)
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001248, A001221, A001222, A006881, A030078, A088381, A088383, A000005.

Cf. A060687, A118914, A323014, A323023, A325249.

Programs

Haskell

a080257 n = a080257_list !! (n-1)
a080257_list = m a024619_list a033942_list where
   m xs'@(x:xs) ys'@(y:ys) | x < y  = x : m xs ys'
                           | x == y = x : m xs ys
                           | x > y  = y : m xs' ys
-- Reinhard Zumkeller, Apr 02 2012

Mathematica

Select[Range[100],PrimeNu[#]>1||PrimeOmega[#]>2&] (* Harvey P. Dale, Jul 23 2013 *)

PARI

is(n)=omega(n)>1 || isprimepower(n)>2

PARI

is(n)=my(k=isprimepower(n)); if(k, k>2, !isprime(n)) \\ Charles R Greathouse IV, Jan 23 2025

Formula

a(n) = n + O(n/log n). - Charles R Greathouse IV, Sep 14 2015

Extensions

Definition clarified by Harvey P. Dale, Jul 23 2013