A167175 -id:A167175 - cp's OEIS frontend

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

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)).

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

A192690 Nonprime numbers with a nonprime number of nonprime divisors.

Original entry on oeis.org

1, 12, 16, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 171, 172, 175, 176, 180, 184, 188

Offset: 1

Juri-Stepan Gerasimov, Oct 15 2011

nonn

			For example, 12 is composite and it has 6 divisors: 1, 2, 3, 4, 6, 12. Of these, 4 are not prime: 1, 4, 6, 12. Since 4 is not prime either, 12 is in the sequence.

Cf. A139118, A141468, A167175.

Cf. A018252. - Omar E. Pol, Oct 20 2011

NonPrimeDivisors[n_] := Length[Select[Divisors[n], ! PrimeQ[#] &]]; Select[Range[200], ! PrimeQ[#] && ! PrimeQ[NonPrimeDivisors[#]] &] (* T. D. Noe, Oct 20 2011 *)

def npd(n: int) -> int:
    return len([d for d in divisors(n) if not is_prime(d)])
def isA192690(n: int) -> bool:
    return not (is_prime(n) or is_prime(npd(n)))
A192690List = lambda b: [n for n in range(1, b) if isA192690(n)]
print(A192690List(189))  # Peter Luschny, Apr 22 2025

A171995 Twin natural nonprimes with nonprime number of prime factors.

Original entry on oeis.org

1, 625, 1715, 1925, 2125, 2875, 2975, 3575, 3773, 4165, 4225, 4235, 4625, 5125, 5375, 5425, 5635, 5831, 5875, 5915, 6517, 6625, 7007, 7225, 7315, 7375, 7595, 7625, 7735, 7889, 7975, 8075, 8225, 8281, 8855, 8875, 9025, 9425, 9775, 9875, 10045, 10075

Offset: 1

Juri-Stepan Gerasimov, Jan 21 2010

Intersection of A171697 and A167175. - R. J. Mathar, Mar 29 2010

			a(1)=1(without prime factors), a(2)=625(=5*5*5*5), a(3)=1715(=5*7*7*7).

Cf. A167175, A171697.

isA171995(n)=n>0&!isprime(n)&((n%6==1&!isprime(n-2))||(n%6==5&!isprime(n+2)))&!isprime(bigomega(n))

Edited and extended by Charles R Greathouse IV, Mar 29 2010

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A323305 Number of divisors of the number of prime factors of n 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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A192690 Nonprime numbers with a nonprime number of nonprime divisors.

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A171995 Twin natural nonprimes with nonprime number of prime factors.

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