A088383 -id:A088383 - cp's OEIS frontend

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

Reinhard Zumkeller, Feb 10 2003

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

			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)

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

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

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

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

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

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

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

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

Definition clarified by Harvey P. Dale, Jul 23 2013

A088381 Numbers greater than the cube of their smallest prime factor.

Original entry on oeis.org

10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 90, 92, 93, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112

Offset: 1

Reinhard Zumkeller, Sep 28 2003

nonn

a(n) > A020639(a(n))^3 = A088378(a(n)); complement of A088380;

a(n) > A088380(k) for n <= 28, a(n) < A088380(k) for n > 28.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A080257, A088378, A088380, A088383.
Cf. A020639, A138511 (subsequence).
Positions of numbers greater than 3 in A307908.

a088381 n = a088381_list !! (n-1)
a088381_list = filter f [1..] where
                      f x = p ^ 2 < div x p  where p = a020639 x
-- Reinhard Zumkeller, Jan 08 2015

filter:= n -> n > min(numtheory:-factorset(n))^3:
select(filter, [$2..200]); # Robert Israel, Aug 11 2020

isok(n) = n > factor(n)[1,1]^3; \\ Michel Marcus, Jan 08 2015

A088382 Numbers not exceeding the 4th power of their smallest prime factor.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, 127, 131, 133

Offset: 1

Reinhard Zumkeller, Sep 28 2003

nonn

a(n) <= A020639(a(n))^4 = A088379(a(n)); complement of A088383;

a(n) < A088383(k) for n <= 67, a(n) > A088383(k) for n > 67.

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

Cf. A088380, A000430, A020639, A088379, A088383.

Positions of numbers less than 5 in A307908.

a088382 n = a088382_list !! (n-1)
a088382_list = [x | x <- [1..], x <= a020639 x ^ 4]
-- Reinhard Zumkeller, Feb 06 2015

Select[Range[200],#<=FactorInteger[#][[1,1]]^4&] (* Harvey P. Dale, Jan 25 2015 *)

cp's OEIS Frontend

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

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A088381 Numbers greater than the cube of their smallest prime factor.

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A088382 Numbers not exceeding the 4th power of their smallest prime factor.

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Haskell

Mathematica