A134333 Numbers n whose number of prime factors (counted with multiplicity) is a prime factor of n.

4, 6, 10, 12, 14, 18, 22, 26, 27, 30, 34, 38, 42, 45, 46, 58, 62, 63, 66, 74, 75, 78, 80, 82, 86, 94, 99, 102, 105, 106, 114, 117, 118, 120, 122, 134, 138, 142, 146, 147, 153, 158, 165, 166, 171, 174, 178, 180, 186, 194, 195, 200, 202, 206, 207, 214, 218, 222, 226

Offset: 1

Author

Hieronymus Fischer, Oct 23 2007

Keywords

• nonn

Examples

			a(1) = 4, since 4 has 2 prime factors and 2 is a prime factor of 4.
a(4) = 12, since 12 = 2*2*3 has 3 prime factors, and 3 is a prime factor of 12.
a(21) = 75, since 75 = 3*3*5 has 3 prime factors. and 3 is a prime factor of 75.
9 = 3*3 is not a term, since the number of prime factors (=2) is not a divisor of 9.
28 = 2*2*7 is not a term, since the number of prime factors (=3) is not a divisor of 28.
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A001222, A100118, A046363, A133620, A133621, A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134334, A134344, A134376, A063989.

Programs

Mathematica

fQ[n_] := Module[{d = Total[Transpose[FactorInteger[n]][[2]]]}, PrimeQ[d] && Mod[n, d] == 0]; Select[Range[2, 226], fQ] (* T. D. Noe, Apr 05 2013 *)

PARI

a(n)=my(t=bigomega(n)); n%t==0 && isprime(t) \\ Charles R Greathouse IV, Sep 14 2015

Formula

a(n) << n log n/(log log n)^k for any fixed k. - Charles R Greathouse IV, Sep 14 2015

Extensions

Sequence definition corrected and examples added by Hieronymus Fischer, Apr 05 2013