A067340 Numbers k such that (number of distinct prime factors of k) divides (number of prime factors of k).

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 49, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91

Offset: 1

Author

Labos Elemer, Jan 16 2002

Keywords

• nonn

Comments

From Peter Luschny, Jul 19 2023: (Start)

If the name means 'Numbers k such that (number of prime factors of k) is divisible by the (number of distinct prime factors of k)', then 1 has to be prepended to the data since A001221(1) = A001222(1) = 0 and 0 is divisible by 0.

Note that the expression 'A001222(k)/A001221(k)' is read as 'the quotient of A001222(k) and A001221(k)' and is not defined in the case k = 1 because A001221(1) = 0. On the other hand, the expression 'A001221(k) | A001222(k)' is read as 'A001221(k) divides A001222(k)' and is well defined also if k = 1 and has the value 'True'. (End)

Examples

			Primes and prime powers are included in this sequence. Another example: 24, since A001222(24)/A001222(24) = 4/2 = 2.
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A001221, A001222, A245080.

Programs

Mathematica

ff[x_] := Flatten[FactorInteger[x]]; f1[x_] := Length[FactorInteger[x]]; f2[x_] := Apply[Plus, Table[Part[ff[x], 2*w], {w, 1, f1[x]}]]; Do[s=f2[n]/f1[n]; If[IntegerQ[s], Print[n]], {n, 2, 256}]
Select[Range[2,91],Divisible[PrimeOmega[#],PrimeNu[#]]&] (* Ivan N. Ianakiev, Dec 07 2015 *)

PARI

v=[]; for(n=2,100,if(denominator(bigomega(n)/omega(n)) == 1,v=concat(v,n))); v

PARI

is(n)=my(f=factor(n)[,2]); #f && vecsum(f)%#f==0 \\ Charles R Greathouse IV, Oct 15 2015

SageMath

def dpf(n): return sloane.A001221(n)
def tpf(n): return sloane.A001222(n)
a = [k for k in range(1, 92) if ZZ(dpf(k)).divides(tpf(k))]
print(a) # Peter Luschny, Jul 19 2023

Formula

A001222(k)/A001221(k) is an integer.