A067028 -id:A067028 - cp's OEIS frontend

A116660 Integers in both sequences A075658 and A067028.

56, 60, 84, 90, 104, 126, 132, 135, 140, 150, 152, 156, 184, 189, 196, 198, 204, 220, 224, 225, 228, 234, 240, 248, 260, 276, 294, 296, 297, 306, 308, 315, 330, 336, 340, 342, 344, 348, 350, 351, 360, 364, 372, 375, 376, 380, 414, 416, 424, 441, 444, 459

Offset: 1

Leroy Quet, Feb 21 2006

nonn

It could be argued that 1 should also be included in the sequence, if 0 is considered to be a composite, since the number of prime divisors (counted with multiplicity) of 1 and the sum of 1's distinct prime divisors are both 0.

			60 = 2^2 *3^1 *5^1. Both the number of prime divisors (counted with multiplicity), 2+1+1 = 4 and the sum of the distinct prime divisors, 2+3+5 = 10, are composite. So 60 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A075658, A067028, A116661.

Select[Range[460], CompositeQ[Plus @@ (f = FactorInteger[#])[[;; , 1]]] && CompositeQ[Plus @@ f[[;; , 2]]] &] (* Amiram Eldar, Nov 14 2019 *)

More terms from R. J. Mathar, Aug 31 2007

A323305 Number of divisors of the number of prime factors of n counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 4, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 3, 2, 1, 3, 2, 2, 2

Offset: 1

Gus Wiseman, Jan 13 2019

nonn

a(1) = 1 by convention.

First differs from A036430 at a(64) = 4, A036430(64) = 3.

Positions of 1's are 1 and the prime numbers A008578.
Positions of 2's are A063989.
Cf. A000005, A001222, A067028, A167175, A323295, A323350.

Mathematica
```
Array[Length@*Divisors@*PrimeOmega,100]
```

a(n) = if (n==1, 1, numdiv(bigomega(n))); \\ Michel Marcus, Jan 13 2019

a(n) = A000005(A001222(n)).

A110893 Numbers with a semiprime number of prime divisors (counted with multiplicity).

Original entry on oeis.org

16, 24, 36, 40, 54, 56, 60, 64, 81, 84, 88, 90, 96, 100, 104, 126, 132, 135, 136, 140, 144, 150, 152, 156, 160, 184, 189, 196, 198, 204, 210, 216, 220, 224, 225, 228, 232, 234, 240, 248, 250, 260, 276, 294, 296, 297, 306, 308, 315, 324, 328, 330, 336, 340, 342

Offset: 1

Jonathan Vos Post, Sep 20 2005

Below 256 = 2^8 this is identical to A067028 (Numbers with a composite number of prime factors, counted with multiplicity).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A001358, A014613, A046306, A046312, A046314, A063989, A067028, A069276.

is(n)=n>9 && bigomega(bigomega(n))==2 \\ Charles R Greathouse IV, Jan 31 2017

a(n) such that A001222(a(n)) is an element of A001358. a(n) such that bigomega(a(n)) is an element of A001358. Union[4-almost primes(A014613), 6-almost primes(A046306), 9-almost primes(A046312), 10-almost primes(A046314), 14-almost primes(A069275), 15-almost primes(A069276), 21-almost primes, 22-almost primes, 25-almost primes, 26-almost primes, ...]

A323350 Nonprime numbers > 1 whose number of prime factors counted with multiplicity is a perfect square.

Original entry on oeis.org

16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152, 156, 184, 189, 196, 198, 204, 210, 220, 225, 228, 232, 234, 248, 250, 260, 276, 294, 296, 297, 306, 308, 315, 328, 330, 340, 342, 344, 348, 350, 351, 364, 372, 375, 376

Offset: 1

Gus Wiseman, Jan 15 2019

nonn

First differs from A014613 in having 512.

			360 = 2*2*2*3*3*5 has 6 prime factors, and 6 is not a perfect square, so 360 does not belong to the sequence.
2160 = 2*2*2*2*3*3*3*5 has 8 prime factors, and 8 is not a perfect square, so 2160 does not belong to the sequence.
10800 = 2*2*2*2*3*3*3*5*5 has 9 prime factors, and 9 is a perfect square, so 10800 belongs to the sequence.

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

Cf. A000005, A001222, A063989, A067028, A067340, A070175, A071625, A118914, A167175, A323305.

filter:= proc(n) local t;
  t:= numtheory:-bigomega(n);
  t > 1 and issqr(t)
end proc:
select(filter, [$4..1000]); # Robert Israel, Jan 15 2019

Select[Range[100],#>1&&!PrimeQ[#]&&IntegerQ[Sqrt[PrimeOmega[#]]]&]

isok(n) = (n>1) && !isprime(n) && issquare(bigomega(n)); \\ Michel Marcus, Jan 15 2019

cp's OEIS Frontend

A116660 Integers in both sequences A075658 and A067028.

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A323305 Number of divisors of the number of prime factors of n counted with multiplicity.

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A110893 Numbers with a semiprime number of prime divisors (counted with multiplicity).

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A323350 Nonprime numbers > 1 whose number of prime factors counted with multiplicity is a perfect square.

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