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

%I A067340 #26 Jul 19 2023 10:19:17
%S A067340 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,
%T A067340 31,32,33,34,35,36,37,38,39,40,41,42,43,46,47,49,51,53,54,55,56,57,58,
%U A067340 59,61,62,64,65,66,67,69,70,71,73,74,77,78,79,81,82,83,85,86,87,88,89,91
%N A067340 Numbers k such that (number of distinct prime factors of k) divides (number of prime factors of k).
%C A067340 From _Peter Luschny_, Jul 19 2023: (Start)
%C A067340 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.
%C A067340 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)
%H A067340 Stanislav Sykora, <a href="/A067340/b067340.txt">Table of n, a(n) for n = 1..20000</a>
%F A067340 A001222(k)/A001221(k) is an integer.
%e A067340 Primes and prime powers are included in this sequence. Another example: 24, since A001222(24)/A001222(24) = 4/2 = 2.
%t A067340 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}]
%t A067340 Select[Range[2,91],Divisible[PrimeOmega[#],PrimeNu[#]]&] (* _Ivan N. Ianakiev_, Dec 07 2015 *)
%o A067340 (PARI) v=[]; for(n=2,100,if(denominator(bigomega(n)/omega(n)) == 1,v=concat(v,n))); v
%o A067340 (PARI) is(n)=my(f=factor(n)[,2]); #f && vecsum(f)%#f==0 \\ _Charles R Greathouse IV_, Oct 15 2015
%o A067340 (SageMath)
%o A067340 def dpf(n): return sloane.A001221(n)
%o A067340 def tpf(n): return sloane.A001222(n)
%o A067340 a = [k for k in range(1, 92) if ZZ(dpf(k)).divides(tpf(k))]
%o A067340 print(a) # _Peter Luschny_, Jul 19 2023
%Y A067340 Cf. A001221, A001222, A245080.
%K A067340 nonn
%O A067340 1,1
%A A067340 _Labos Elemer_, Jan 16 2002